It's just ego. BTW, you don't actually have to refuse each draw offer. That can indeed become tiresome. Just make your next move and enjoy your opponent's discomfiture...Oh wait, that sounds like ego too, doesn't it? LOL
Players Asking Draws in Lost Positions

I can't tell if they're low rent or have no idea what they're doing. I had mate in one and my opponent asked for a draw.

Well it doesn't hurt to ask for things It's totally up to you if you want to accept anyways. I can imagine a situation where someone realizes they made a serious blunder, and they ask for a draw on the chance you did not notice.
I haven't asked for one yet, or have had someone ask for one, but if someone asked, I'd probably just give them the draw.
I’ve been playing on chess.com for quite a while and have noticed recently that a large number of players are making draw offers after they blunder. In a blitz game that I played my opponent blundered a knight and then offered a draw, to which I obviously refused. He commented “fair play?” and “fair play no exist?” to me multiple times of which I ignored. At first I thought it was just an oddity I observed until in a daily game of mine in an almost identical position my opponent dropped another knight. While I am grateful he did not make staunch claims about fair play in horrendous English I am still somewhat surprised by the draw he offered - what is going on with this?