...The political reasons explain why Hitler did not really want to invade Britain. The practical reasons, however, are even more important - they are why he could not hope to successfully invade Britain even if he wanted to. There are four factors which are either absolutely necessary, or extremely important, to an amphibious invasion, none of which Germany possessed. They are the ability to transport a sufficient invasion force to the landing zone, the ability to supply that force once it has landed, air superiority, and the ability to protect the invasion force and its supply lines from enemy naval attack. Of course you must also possess an army to send across, but Germany had no shortage of armies.
First, the ability to transport an invasion force across the English Channel requires landing craft, and lots of them. Germany had very few, and they were of very poor quality. Plans for Sealion involved using Rhine river barges for transport across the Channel. The problem with these was that they were not designed for use in the ocean, and would swamp if exposed to anything but the very mildest weather (or if a large ship passed close to them at high speed). Even if this were not a problem, there simply were not very many of them. The Germans estimated that they had sufficient craft to ship across an invasion force of at most ten divisions. Ten infantry divisions, because without proper landing craft, heavy weapons such as artillery and tanks could only be transported with extreme difficulty. The "best" scheme the Germans could come up with was shipping across tanks on barges, one to a barge, and having them shoot off the front end of the barge so that they could exit it at the landing zone - a method virtually guaranteed to lead to a high rate of failures of tanks attempting to land.
Without a major addition to their landing craft fleet, which would take a great deal of time to build and would be very obvious to the rest of the world, the Germans could not hope to send across more than ten infantry divisions with almost no heavy weapons in support. A force of this size would be slaughtered by Britain's defenders, which included many divisions of soldiers evacuated from France and equipped as infantry, enough armored forces to outnumber anything the Germans could bring across, tens of thousands of Home Guard militia, and several fully equipped divisions of reinforcements from Canada. In event of an invasion, the British government was fully prepared to use all means at its disposal to stop it, such as poison gas attacks and flooding the English Channel with burning oil. Poison gas could be used by the Germans as well, but they would need time to prepare countermeasures and to use their own gas. The British would gain a short-term advantage by being the ones to introduce gas, and a short-term advantage is all they would need to crush a fledgling invasion attempt.
Various schemes proposed to get around the outnumbered nature of potential German attackers have been proposed, but none would have been workable. Using paratroopers wouldn't work - even if the Germans had not lost most of their paratroopers in the invasion of Crete, and even if their slow, extremely vulnerable transports somehow got past the RAF, Britain was far too large and well-defended for Germany's paratrooper force to make any real difference. They excelled in attacks on pinpoint and isolated targets, such as Eban Emael and Crete, but jumping into a large area with many divisions of infantry and tens of thousands of militiamen, they would be slaughtered before they got a chance to do much of anything. For the Germans to use gas first would not work, because within a short period of time the military forces of both sides would use countermeasures. The main harm of countermeasures is that they slow an army down - of little harm to the British defenders, who can stay in their fortifications, but much more harmful to an invasion force attempting to seize beacheads and take territory. Having the Germans succeed at capturing more, or all, of the British forces evacuated from Dunkirk would also be insufficient. A little-known fact is that after being evacuated from Dunkirk (minus their equipment), most British soldiers were sent right back to France where they fought on until the final pullout. The British lost their heavy equipment at Dunkirk anyway, a fact which can hardly be made worse, and even without the evacuated men the British had more than enough infantrymen to fight off a German invasion. The Germans were physically incapable of shipping across an invasion force even half the size of the one they would need to have any chance of beginning a successful invasion.
Second, the supply situation. Once you have sent across an invasion force, it needs to be resupplied and reinforced before it is pushed back into the sea. An invasion force can't carry enough supplies on the landing craft to last more than a day or two, has few heavy weapons, and is almost certainly outnumbered by the forces the enemy can bring to bear given enough time. It needs extensive shipments of supplies , especially in the first few days, or it will run out and be annihlated. It needs heavy weapons and armor, or it will be crushed as soon as the enemy has enough time to bring the full force of his own heavy weapons to bear, or at best be unable to expand far from the initial beachead. It also needs large amounts of reinforcements, so that the main body of the army can be brought across and change the battle from a fight to gain and maintain a foothold, into an actual conquest.
Germany's ability to do this was, quite simply, very poor. Germany was never much of a seagoing merchant power, and as such it did not have very many freighters. Those it did have did not have the equipment or the space to be able to transport German tanks, a serious handicap. Much of the invasion's continued supply would rest on using the Rhine river barges... the same ones used to transport the initial landing, many of which would likely have been destroyed in that same landing, and which were at any rate both slow and vulnerable to attack and poor weather conditions (you can choose to invade on a pleasant day, but it's just tough if four days later, your supply fleet is decimated by high waves). The supply line of Sealion would be ridiculously inadequate, and would need to be many times larger to be physically capable of transporting enough supplies for a credible invasion. For Germany to build a substantial invasion fleet, or supply fleet, would require time (at least a year), would be very obvious to the rest of the world, and would divert resources from other programs. Germany was generally at the limits of what Hitler could realistically finance for the war at the time, so building more barges and freighters would mean less tanks, less airplanes, or less warships, any of which would hurt some other part of the war effort somewhere else.
The third factor is air superiority, easily the most famous of them all due to the Battle of Britain. Much too famous, since many people falsely assume that German victory in the Battle of Britain is all that would be necessary to permit a successful invasion. I hope that the rest of my essay demonstrates that it is not all that is necessary, but indeed, Germany does need to establish air superiority - preferably even total air supremacy - over the Channel and the invasion beaches for the invasion to work. The problem is that winning the Battle of Britain doesn't allow them to do this. Even if Britain was not consistently outproducing Germany in aircraft, and even if Britain did not have numerous advantages such as fighting on home turf (meaning that they could usually recover pilots who were shot down, while the Germans could not), and the use of Radar to give early warning of German attacks, that would not be good enough. Even if the British did not have the Spitfire fighter (which was not so important at that point in the war, when Hurricanes were much more numerous, shot down many more aicraft, and were considerably cheaper to build and maintain), even if the Germans had continued their program of bombing RAF airbases instead of turning to attack London (which would hurt the RAF more, but not be nearly enough to turn the tide), even if the British leadership had been different and decided to follow the much-inferior "Big Wing" policy of air defense, this wouldn't be enough.
The reason none of this would be enough is because of what the RAF planned to do if it lost the Battle of Britain. Quite simply, they would withdraw all surviving fighter groups to the north of Britain, out of range of German fighters, where they would be essentially invulnerable to attack. They would wait there until the Germans launched an invasion attempt, whereupon they would immediately fly south en masse to attack, denying the Germans air superiority. So due to this quirk of geography and German fighter range, there is basically no way for the Germans to get air superiority over the invasion (without, say, multiplying the size of their air force by many times - which would, again, require great advance planning and mean taking resources from some other part of the war effort), because the British would withdraw enough aircraft to safety to cause serious problems for an invasion. Something often overlooked about the Battle of Britain is that the British had multiple fighter groups, several of which were based to the north, out of range of attack. These were used as places where the pilots could rest, aircraft could be repaired, et cetera. They were at fairly high strength during the Battle, and thus even total annihlation of the aircraft actually in the fight would leave the RAF with plenty of aircraft in reserve for Sealion.
Withdrawing to the north would indeed leave the south of Britain vulnerable to bombing, but bombing was never decisive in the war even when the Allies launched thousand-bomber raids against poorly defended targets in 1944. In 1940 the Luftwaffe bombers, flying unopposed, would cause a good bit of damage and be very annoying, but they would not seriously impair Britain's ability to carry on the war, or to build up its defenses against German invasion. As such, if the bombing campaign continued unopposed before an invasion, it still would not be sufficient to weaken Britain's defenses (or, actually, prevent them from strengthening) to any great extent. Unopposed bombing would thus be of little to no help in preparing the way for a German invasion.
Last, but definitely not least, is the ability to protect the invasion fleet from naval attack. Something that is often sadly ignored in Sealion scenarios (except the ones that the Germans themselves came up with, one of the main reasons Sealion was never more than a pipe dream to them), is that the invasion fleet and its supplies must cross a body of water known as the English Channel. Water is the domain of the Royal Navy, at the time renowned as the most effective fighting force on the oceans. The forces of the Home Fleet, stationed at or near Britain at all times (usually at Scapa Flow naval base, out of range of German air attack), included at least one aircraft carrier, half a dozen to a dozen or so battleships and battlecruisers, and over a hundred smaller vessels such as destroyers, cruisers, and frigates. Dozens of the lighter vessels were stationed around the southern coast of Britain at any given time. In the event of an invasion, the ships already in the south would cause serious damage to an invasion fleet. Even if the invasion came as a complete surprise, within 24 hours the majority of the home fleet would be sitting in the middle of the English Channel, sinking everything that came within sight. They would certainly take losses, from various forms of attack, but this would not have stopped them. With their country about to be invaded, every last ship would be sacrificed if necessary to stop the invasion. It would not be necessary, however, because the Germans hadn't much to throw back at the battleship task forces in the way of its assault.
Most of the German navy was composed of U-boats. Great for commerce raiding, lousy for attacking well defended convoys, especially in 1940. Even lousier for attacking entire fleets of warships. Not to mention the fact that in the English Channel, in an area packed with destroyers and with very little room to manouver, using U-boats would be nothing short of suicide because they would have nowhere to hide. The German surface navy, at its height, never consisted of more than one battleship, a few battlecruisers and "pocket battleships", and ten to twenty lighter vessels. Thus, the Kreigsmarine at the height of its power was outnumbered between 5 to 1 and 10 to 1 by the Home Fleet. Not good odds if you have to not just fight a force of ships, but prevent them from so much as surviving for a day or two in the middle of the English Channel to sink your barges. At the time at which Sealion would have been likely to actually occur, the odds were even worse. The battleship Bismark was not yet finished, and the Germans had just successfully invaded Norway. This "success" left almost all of their fleet either destroyed, or having serious damage repaired. They had a few cruisers and destroyers to throw against the Royal Navy.
The German fleet may thus be entirely discounted from the question of how the Germans could prevent significant elements of the British fleet from sailing into the channel and sinking their invasion's supply line. The U-boats were useless in such a situation, and even if the surface fleet was not temporarily out of service due to Norway, it was not anywhere near large enough. Again we run into the old problem that if it is to be large enough, the building plan has to be started in advance (for capital ships, at least five years in advance!), and will take away substantial resources from the rest of the war effort. Battleships require a lot of steel to build. It doesn't work to postulate that the U-boats will starve Britain into surrender, either. First, it would be very hard to build more - they required precision manufacturing of many components, skilled workers to build them, and skilled officers to crew them. All of these were in quite short supply. Second, Britain is weakest in the later half of 1940, after that it becomes much stronger quite fast, and within a year is so well defended that there is no chance of a successful invasion. The problem is that this leaves only a year between the start of the war, and the time when Britain is supposed to be starved into being weak enough to allow an invasion. Not nearly enough time for a U-boat force of even twice the size and capability of the one the Germans had in 1940. The U-boat force was most effective later in the war, when it had more U-boats, better U-boats, and bases in France from which to easily reach the Atlantic. It must also be pointed out that building more U-boats detracts from the rest of the war effort, will likely lead to increased antisubmarine countermeasures by the British to compensate, and doesn't really help the problem of those battleships sitting in Scapa Flow deciding to sail down and sink the invasion's supply line.
Last but not least, aircraft - also not enough to stop the Royal Navy. The main problem is that they don't have the range to reach Scapa Flow, so they can't actually attack the British ships until they are already well on their way to where they will do the most harm. The Germans didn't have much capability to attack ships, anyway. They had no dedicated naval attack aircraft, no torpedo bombers, and their pilots lacked both training and experience in naval attacks. This was demonstrated in the Norway campaign, when they achieved a very low success rate against outnumbered, unarmed ships. In contrast, a battleship task force with a full screen of cruisers and destroyers has a tremendous number of antiaircraft batteries with all-around coverage, and can deal tremendous punishment to enemy aircraft. In the Pacific war, even when both sides had extensive antiship capabilities, air battles between American and Japanese fleets often lasted for many attacks over a period of days, with hundreds of aircraft being able to sink many major vessels in a task force, but virtually always leaving many survivors. Even a Royal Navy force with no air support at all could survive for the required few days in the channel under attack by an air force much better against ships than the Luftwaffe.
In fact, the Luftwaffe would have been quite bad against ships. Virtually all of its bombers were level bombers, which drop bombs from high altitude against stationary targets to good effect. Ships, however, can manouver so as to make themselves harder to hit - and level bombers thus become poor choices to use against ships even in the hands of expert pilots (only the Japanese had any real success with them in the war). Dive bombers and torpedo bombers are generally more effective. As mentioned previously, Germany had no torpedo bombers and its only dive bomber was the Stuka. The Stuka was the terror of the skies in the 30s, but by 1940 it was considered slow, vulnerable, and short ranged. Stukas would have suffered horrendous loss rates against the intense air defense of capital ship groups with concentrated destroyer screens. It's also worthy to note that, due to their range and speed, they could only make an absolute maximum of three attacks on Royal Navy elements sailing from Scapa Flow before they reached the channel. Realistically, only one or two. Thus the British fleet elements sailing south to stop the invasion would not experience significant air attack until they were already within range of the invasion fleet and its supply lines.
All these factors together should demonstrate that the Germans had no capability whatsoever to conduct a successful invasion of Britain. They lacked the ability to get a sufficient invasion force to Britain, to supply and reinforce it once it arrived, to protect its supply lines from the Royal Navy, and to establish protective air superiority over the invasion. For any Germany at all like the one we are familiar with, establishing air superiority during the invasion and protecting the supply lines from the Royal Navy are effectively impossible. The British can withdraw aircraft out of range of attack until the invasion, meaning that a huge increase in the Luftwaffe would need to occur to be able to establish air superiority despite these RAF reserves. Both a substantially increased German surface navy and a substantially increased Luftwaffe and Luftwaffe naval attack capability would be necessary to defend the invasion fleet against the Royal Navy. The problem with trying to remedy all of these is that they will need to be done at the expense of the army, leaving it much weaker for the invasion of France that has to happen before you can even get to Britain. Since Germany's aims lie on the continent - especially the primary war aim of Russia - Hitler cannot afford to focus on a strategy that favors amphibious invasions. And if he does, this must start well before the invasion is to occur, so that years in advance, it is plain to Britain that Hitler is persuing a strategy that can have no realistic use except in invading Britain.