This is similar to my answer to this question but I felt that it was also appropriate here.
Livestrong has a great page that illustrates a lot of points and references medical studies to support it.
Some key points:
limits extra muscle motion, or "oscillation" ... [which] reduces the amount of energy you are expending
assist[s] the hamstrings in controlling leg movement at the end of the swing phase ... delay[s] muscle soreness by compressing the muscle tissue enough so that less damage occurs
reduces impact force to muscles by 27 percent
increases the amount of lactate retained in muscles and thus reduces the amount that is released into the blood
Compression from the shorts may protect against a venous thrombosis
Compression shorts may help cool the body by wicking sweat from the body and allowing it to disperse faster
Take a look at the page for more details.
From a personal opinion standpoint, I've found that I personally perform much better (running and soccer are the main events I use compression). The shorts also offer much needed support for the man-specific area.
Additionally, I use compression sleeves for my calves. I suffered from shins splints and the compression sleeves seemed to almost completely alleviate the shin splints when I would run. I've spoken with therapists about this and the reason I was given was because the compression sleeves keep the muscle attached to the bone during running (shin splints are micro tears of the muscle away from the bone).