It's a term that seems to just have crashed in the LoL community. I now seem to see 'This play was so BM' or 'Look at the BM'... It seems to be only with cocky plays that went wrong.
So what is BM?
BM stands for Bad Move/Manner/Mouth.
This term isn't exclusive to LoL and is used in quite a few games such as Hearthstone, Overwatch and just about any competitive game.
It's basically when you are being a jerk unnecessarily to someone. e.g "T-bagging" someone, leaving someone on 1 HP just to kill them anyway, taunting, etc.