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.

