I agree with all the other reasons that the other posters gave. I'm surprised no one mentioned this however.

Personally, I ride geared bikes most of the time. However, there are days when I've got limited time to get in a ride, like an hour or so. If I take a geared bike on an hour ride, I get back feeling like I hardly got a workout. If I take my fixie road bike on an hour ride, I feel like I get a better workout. I would estimate the factor to be about 1.5. That is, for every hour you ride a fixed gear you can get about as much exercise as riding a geared bike 1.5 hours. All that from not coasting.

I guess I could get a similar effect from riding my MTB on the road with knobbies inflated to 30 - 40 psi, but that would be noisier and not as much fun. I like the position on the road bike better. Plus it's so quiet and smooth.

Bonus - when I first started riding fixed I was forced to pedal through turns instead of coasting. This is a good method of improving your cycling technique. Put low gears on and force yourself to improve your spin if you like, too.

In conclusion, in addition to being fun and being one with the bike, it is also a great training tool.