Unlike this questioner, I'm not asking what my phrase means (in case anyone doesn't know and can't guess, it means to talk incessantly).
But I don't know anything at all significant about donkeys' hind legs (apart from the possibility of them being metaphorically talked off).
I doubt we can find an actual origin for the phrase, but perhaps someone can come up with a plausible reason for how it came about, and/or why it continues to be used.
Best Answer
The phrase originates in Ireland. Donkeys (or "asses" or "jackasses" as they are called in other parts of the world) do not naturally sit down on their rear ends. In fact, it is an extraordinary achievement to get one to do it. "Talking the hind legs off a donkey" is a literal translation of the Gaelic, which actually means "making a donkey sit down on its rear end". Thus, when a person can talk the hind legs off a donkey, they can talk so much that they could even bore a donkey into sitting down.