For me that phrase means that someone just keeps on talking, whether or not you want to listen. I wouldn't associate it with being persuasive, as you generally want to get away from someone who'd talk the hind leg off a donkey. I was told that a donkey's hind legs are very strong, and to take or "talk" them off would require a lot of effort. I don't know though.
I don't think the phrase came from Irish, as in most phrases I've heard "asal" would've been translated as "ass", and there's probably a pun in there between "talking" and "taking".
And more to the point, what did the donkey ever do to you that you'd want to go i dtigh an asail
