Draw offered? What?

This is a game I played.  I've been playing for 3 months, so I'm no GM, but I thought I played well.  My opponent offered me a draw after my last move, and then resigned after I declined.  I understand resigning, but why would I accept a draw?  Would love to know what people think.  If you see large errors, I love to know where and how I screwed up!

Merci beaucoup!

People sometimes offer a draw in a totally lost position in hopes of that you'll accept it for whatever reason (even by accident) and thus they'll get a better result than what is inevitable. It's not a very courteous habit, though.

I like 22.Bd5 :)

I don't get it.  I assume it's a joke, because I'm looking and don't understand the tactical value.  If there is really something there, I'd like to know.

I did notice that 21. Qxb8 is a free Knight.

I also noticed that the game starts with the hypothetical I made, which means I don't know how to work the game diagram too well.

Good game. Nice planning you did there.

P.S. Are you French? I know 'Moi' means 'Me'. So does 'Autre' mean 'Opponent'?

No, I'm not French.  I've studied it in college and loved it, though.  Autre is other.  Technically I should use L'autre for The Other, but oh well.

with regard to wafflemasters suggestion of 22. Bd5, one thought is that this would clear out the pawn on e6 (assuming black takes the bishop on d5).  The freed-up space would then allow, say, Qe7, which puts black in a whole lot of trouble.

Being 18 behind is a whole lot of trouble anyways.