Bettas can "blow" their fins. No one has yet found any definative reason why, it seems to vary from fish to fish- though it's usually stress/aggression based. I had one male do it, once- the webbing between the rays of his fins was just... gone... or shredded. It was left floating around the bottom of the tank... ick.
Some also bite their own fins out of nervousness/stress, and leave chunks. You probably won't know for sure what, but check for jagged edges on tank decor, plants, etc- maybe cover your filter intake tube.
These guys sure do keep us on our toes, don't they? Best of luck with him and his recovery
