Because it's fun! That's the short version.
Medic is a class often overlooked by players when choosing a class. You have basically no DPS, no tanking potential and when cornered by any of the other 8 classes, you are in most cases already dead.
So why do we play as this glorious German doctor, who cares so little for his patients, that he supposedly lost his medical licence because he stole the skeleton of an anaesthetised patient?
#1 The healer
Some people like to play medic because they like to help people. They like to know that the other players appreciate them and that what they do, makes a difference. This is a great playstyle to start out with if you have never played Medic (or Team Fortress 2 at all, for that matter).
The healer does just that. He heals. He makes sure his teammates don't go belly-up and die a horrible bullet-and/or explosion-filled death.
This type of Medic is usually great, except for their occasional lack of skill. A mistake that some newer Medics make is just standing still while healing. This makes them an easy target for snipers and spies. And, well, everyone else.
#2 The Battle Medic
Probably the most hated type of Medic. This guy will hardly ever heal other players, and will do whatever he can to do as much damage as possible. Now, playing a Battle Medic can be fun, sure, but it's not really helpful to the rest of the team.
The Battle Medic will wield the Blutsauger with massive overconfidence and spray away on any other class he sees. This means that the Battle Medic tends to die. A lot.
The people who play Battle Medic does it because it's fun. They like the difference in playstyle from the offensive classes; the fact that pretty much every other class is better than Medic when it comes to offence, makes pulling off a 1v1 victory as a Battle Medic that much more rewarding.
#3 The God Complex
This type of Medic, is a bit like the Healer. They like to heal and see that they make a difference, though not in order to help the team. They heal to assist in the vicious slaughter of the opposing team. The difference that like to make is to be able to determine who lives and who dies. These types of Medics love to be the deciding factor in helping a Heavy destroy an entire sentry nest by using his Übercharge at just the right moment.
This type of Medic can be really great to play with, if they have the sense to heal other than just whoever they pocket.
I fall into category #3 myself. I just love being able to choose a target and help them achieve greatness, all along completely eradicating the enemy team.
Well, that's it. My 3 general reasons why people play Medic. Feel free to leave a comment on how you play medic or if you have any additional playstyles I haven't mentioned here.
~
Decadence