There are several theories of the origin of the saying. Here is one of the most persistent explanations.

Any time a person or an object moves through or past a drop, they are said to be "Breaking Curtain." The thin curtains that mask the wings are referred to as "legs." Therefore, to enter from the wings, an actor must first "Break a leg."

Back in the day, particularly in the era of vaudeville, variety shows consisting of assorted talent acts were fairly common. Often the crowd's reaction led to a person either being allowed to finish their act, or forced off the stage. But, even a few minutes in the limelight was good exposure for an actor.

Ergo, even if their performance wasn't great, or if the actor was shot down before he got a chance to perform, he still had managed to get his name and face out there by breaking the legs.