I got curious and googled it and it appears to have been used at least since the 70s but it was maybe restricted more to people involved with sports, especially basketball.
This is how I remember it starting. Basketball players admitting to personal fouls. I absolutely detest the term used to acknoweledge a mistake. It sounds so ignorant to me. What is so hard about saying "my fault" or "my mistake" or just sorry.
What is so hard about it? Using alternate phrases adds flavor and nuance to our language. It is limiting language that is "ignorant" not expanding it.
See, I don't know that I agree with this entirely. I personally like to flavour my speech with all sorts of 'mad' sayings that I've picked up from international friends or my students. Playing with language is one of my personal delights.
However in this *specific* case I am finding that a significant portion of the teenagers that I teach ONLY know "my bad". They would never, ever say "my fault" or another alternative because they just don't know they are available, or if they do they see them as archaic terms used only by teachers. Something about that bothers me. It feels too much like "ungood" I guess.