Did you ever play Civ IV without BTS? Because that would explain your confusion.
City investigation and visibility are now passive espionage missions. You don't have to do anything other than have sufficient espionage points against a target. Go into your espionage adviser screen. Select a leader. You are presented with a list of his cities. Select one. Look at the box to the right. It has, among other things, the phrases "City Visibility" and "Investigate City" with numbers next to them. If you have banked >= that number of EP against that leader, you gain that benefit. City visibility (which used to be a function of religious shrines) removes fog of war on the city tile and all adjacent tiles, and investigate city (which used to be an active spy mission) lets you open the city screen like it was one of your own, though you can't change anything.
Easy-peasy. Religion is still involved, though it is no longer the only factor. You get an espionage discount for missions (including the passive ones) if the target city has your state religion present, but it is not their state religion - assistance from disloyal, oppressed minorities. You get an additional discount if you control the holy city of that religion. The combined discount (40%) makes the cost of "city visibility" very low and easily achievable, and "investigate city" is much more attainable against your primary EP target.
