It's really hard to say without context. The first thought in my mind was a horse's bit ("bocado", ie. that part of the bridle which goes in the horse's mouth). So if you "commit to the bit", it would mean someone else has given you a task or role, probably one that you wouldn't normally take, and now you're doing the task (or playing the role) very well, to the point that you seem to be enjoying it. You really wouldn't come across this phrase often. We do have other phrases relating to a horse's bit: http://en.wikipedia.org/wiki/Bit_(horse)#Idiomatic_usage

Without knowing where the line comes from, it sounds like the 'bit' in this case is a comedy bit, like a sketch or a joke. So the person is admiring the other person for committing to the joke. In other words, they are playing their part even when it's uncomfortable or difficult for them in order to make the joke funny.