The term "My Bag" was originally slang for taking responsibility for one's own actions. Somewhere in the early 90s people started saying "My bad" either because they wanted to change the verbage of the original, or because they just plain didn't understand the original meaning. "My Bag" is slang for "Baggage",ie: You take your own problems, put them in a bag and carry it on your own back. "It's on me, my fault, my problem. "My Bag" excepts and takes the burden, "My Bad" just dismisses it.
"Joe, do you need help paying for your rent this month? I know you haven't been working and would really like to help you out." said Nick. "No thanks man, that's My Bag." said Joe. "Oh, My Bad, I just thought you could use the help dude."