What does it mean when a Betta's tail starts to fall off?

This could be one of many reasons, the most frequent culprit would be other fish, for instance gourami and betta come from the same family of fish, more often than not they don't get on also there are other species of fish such as rosy and tiger barbs that are very "nippy" and like to go around the tank nibbling at long flowing fins, i would need more info to properly diagnose but that is my first thought.

If you have a spare small tank i would suggest taking you little fighter out and putting him on his own, they live mainly in mangroves so you could make the spare/new tank/bowl to look like this, if you do then add some melafix to the tank, melafix will encourage the regrowth of fins.