Its a trigger that fires the gun when you let it go (release) rather than when you pull it. There are several theories to support their use.
The first is that they will be more reliable if you flinch. It works for me, but I still flinch once in a while with the release, mostly early in the season. I can barely pull the trigger in sporting clays if I start with a mounted gun. On doubles targets, I have to take the gun off my shoulder between shots or I'll jerk like an epileptic.
The second reason is that it is a more natural action to release a trigger than to pull one, and results in less muscle tension. Put as simply as I can state it, with a release, you bring the barrel to the bird and pretend to point the finger at the target. Seems easier than jerking the finger backward.
Don't start using a R/T unless you really have to. I believe many people use them soley because "so and so uses one so they must be better". Not on your life. If you don't need it don't start. They are also now illegal in Olympic Trap.