Saying that you "tapping out" is a way to signal that you've said all you wanted to say on a particular topic, and that you're ending your participation in a conversation lest you start saying things that are less worthwhile. The LessWrong tradition was started by Rain in this comment.
Tapping out doesn't mean accepting defeat or claiming victory, and it doesn't mean you get the last word. It just means that you don't expect your further comments in a thread to be worthwhile, because you've already made all the points you wanted to, or because you find yourself getting too emotionally invested, or for any other reason you find suitable.
The analogy comes from the practice of "tapping out" from a sparring match when one is tired, or at risk of injury, or has simply had one's fill.