Watching an MLB Gameday live lookin today (7/6, NYM@CIN), I noticed Brayan Pena of the Reds batting with a leg brace. He singled, and removed the brace at first brace?
Why might someone do this?
Assuming you're talking about the 'armor' some batters wear (not a purely supportive brace made out of terry cloth or similar), the leg brace is usually to shield the batter from a ball striking his leg, often caused by a foul off of a ball in on the hands. Those balls can come with substantial force directly down into a player's ankle or calf (or higher occasionally), causing potential injury.
The batter would remove the brace once he becomes a runner (and can do so without risk of being out), since it hinders his ability to run.
Odds are he was injured at some point and decided to wear the brace to protect his leg. I believe Pena had a hamstring injury last month, perhaps that is vulnerable to re-injury due to this sort of thing.
It's also possible that the brace is there to prevent an injury that might be caused by the twisting motion of hitting (or trying to hit) a baseball. Running the bases is a different motion and one that's less likely to involve that much torque on a joint.