I think it might work the other way around, actually - the winner offers the loser the chance to just give up the game, knowing there’s no hope of winning. Sort of a mercy killing, I guess
Remember, the quote started with the person’s name - say John Smith and he was the one with just the king. So, assuming your translation is correct (and I believe it is) it would read
John Smith offers a remise.
He doesn’t stand a chance, so that’s why I think he was being funny. What’s more, a couple of moves later when the situation was even more bleak the very same quote was repeated.
And by the way, either side can make an offer. Usually it’s an offer to draw - usually, I think, by the person who fears he might have a losing position.