1) They're the only way to get air support for the initial attack on an enemy continent -- so their obviously good in that situation. After you have a city, you can base a whole air force in that one city, which will probably be able to provide air support for your attacking army. So the role of the carrier shifts.
2) I generally use them to keep an eye on how my enemy is attempting to bring reinforcements to the front. A couple fully loaded carriers will provide excellent recon, and can ensure that anything that doesn't have an SAM escort arrives to the front at about 50% health. The recon is also helpful to ensure that the enemy doesn't circle via ocean to your rear, and attack your flank.
A couple of games ago, I went to war against England (on a different continent), and Cyrus joined in on England's side. Initially, I didn't worry to much (Cyrus only had 2 port cities, each of which I had a battleship parked outside of, and he didn't have any direct borders connecting England). Eventually though, Cyrus marched his army through some Open Borders countries and right up to the front.
What might have happened, is that his army might have attacked, full strength, against my wounded army that had just taken a 2nd English city. That would probably have wiped out my invasion force, and after Cyrus' army rested, it might have been able to retake London, and succeed in kicking me back off the continent.
But my forward deployed carriers alerted me to Cyrus' advance a couple of turns before he arrived, and was able to do some harassment to damage some of his troops. So what DID happen was his army attacked my full strength army on a forested hill (instead of taking a 2nd English city, I paused and positioned my army for a better battlefield). Cyrus lost almost everything he had, while most of my army survived and was able to do an organized retreat to London, where they healed and then recommenced the march across England. The carriers were key in ensuring that my invasion was just slowed for a few turns by Cyrus' army, rather than wiped off the continent.
All that being said, Carriers are still not nearly as important in the game as they are IRL. I think they're more important than in CivI or CivII, but they're still not comparable to real life.