To keep water from the road from splashing up on you and the bike. Some people consider them an absolute requirement for commuting, others not as much. I use them during the winter and wetter parts of the year. If you ride on wet roads without a rear fender or at least something between your back and the rear wheel, you're likely to end up with a dirty looking stripe on your back.

There are fenders that can be put on almost any bike. Lots of bikes (including many road bikes) have mounts specifically for them and that makes it a lot easier.