The key to a flavorful steak is its juices. Once you understand this, the reason to rest the meat becomes very apparent.
When a steak comes right off the grill, the temperature on the outside of the meat is very high. At high temperatures the muscle fibers cannot hold as much liquid, so the meat juices are forced into the center of the steak. If you cut the steak open right after it comes off the grill, all the juices will flow out of the meat. However, if you wait, the temperature of the outside of the meat goes down, and its ability to retain liquid increases. The juices are redistributed throughout the meat. When you cut a rested piece of meat, no liquid will flow out. This means all that wonderful, flavorful juice is still in the steak, so when you eat the steak, it will taste wonderful.
The optimal resting time is dependent on the thickness of the meat. For steaks, resting at least for 10 minutes should be enough.