Bettas sometimes eat their own tails. There's no clear reason as to why they do this. Some people think it's stress from water conditions/seeing other males/etc., and some think it's genetic. He may never stop doing it, but the best way to help it grow back is clean water with some salt and melafix.
right on for the treatment.
but.... I wouldn't jump to tail eating that quickly. Personally I have never had this problem with one of my bettas..... but both of my gf's bettas have ended up with a little bit of tail damage, then a bit more after a week..... but it wasn't progressive like fin rot.
the 2 factors different from her fish and mine:
1) hers have gone for 3 days without being fed (possible they nipped their own tail, but the damage didnt seem consistent with that). she comes back and the tails are gone.
2) she has high ammonia in her tap water.
..... i was thinking ammonia burns at first. both tanks were new and not very well cycled. Her established and healthy fish now goes for long weekends w/o food and has been growing fins back like it's his job.
One now has clearly come down with fin rot, as the rips have come towards his body, but i diagnosed the fins personally and told her that it was from the fins being caught in the filter (which is now lined with nylon). It's SLOWLY been getting worse without the normal signs of fin rot. After being treated with Maracyn 2 for 5 days in a hospital tank, he has no more fin damage and is apparently having some re-growth.
My opinion. remove all possible factors for the problem. Get an easily accessible 1 gallon tank and fill it with artesian water (NOT DISTILLED). Condition the water accordingly. NO plastic or real plants. NO meds or treatments. observe very closely for a few hours on and off to see how the fish is acting (health wise). Check out the tail for at least a day. If you suspect any sort of finrot, treat accordingly.
My gf's fish (with almost no fins

) is now re-growing and acting as usually after the antibiotic treatment. Moral of the story, don't underestimate the problem. Best to be proactive.