Some bettas chase and bite their tails (like a dog), he probably did this, or it could have gotten caught in the intake tube (if you have filtration) when he swam close.
The brown could be the onset of fin rot though...you might wanna get some fin rot medications (of your choice. My preference is Salt and Stress Coat for minor cases and Melafix for advanced cases. Unless you have, or can get, some Indian Almond leaf ..which I would reccomend above all else if possible to acquire).
I would stay away from antibiotics unless it gets really bad (like half his tail is gone).