gxf3 e.p.
How isn't this mate?
The pawn in g4 can tak the "mating" pawn. It can only do this in the move just after the white pawn moves two ahead. I'm not sure of the english terminology, but in polish it translates roughly as taking during a flyby.
Edit: beaten to the point.

gxf3 pawn en passe. If you didnt know about it, of course you would feel jipped. The real question will be if your opponnent knows about it.

gxf3 e.p. See www.chess.com/chessopedia/view/en-passant.

I've got a question about this situation.
Isn't the pawn capture mandatory? This is not checkmate and black must get out of check. If black doesn't resign then white doesn't get to claim the win without a checkmate. Black must play or eventually lose on time trying to figure out a legal move.

Everyone already said it - but the en passant pawn capture prevents mate. They initiated the rule when pawns BEGAN making the double move at the beginning to speed up the game - players worried that the ability of making a passed pawn so easily could unbalance things so they made a new rule - that pawns could capture a pawn that had just made a double move AS IF it had only moved one square forward but only straight after its first move. It looks kind of odd, and is the only move which involved capturning a piece on a square its not even on but that is the rule.
gxf3 stops mate and seems to leave black with an ok position.

I've got a question about this situation.
Isn't the pawn capture mandatory? This is not checkmate and black must get out of check. If black doesn't resign then white doesn't get to claim the win without a checkmate. Black must play or eventually lose on time trying to figure out a legal move.
Yes, correctly reasoned... but I see there's a faint ethical dilemma here -- is that what you're getting at? Suppose it is Black, not White, asking the question here...and it's an on-going CC game played according to Chess.com rules -- it's clear that Black could refer to printed rules and legally figure out his dilemma, but here, he'd be asking for help from other people to figure out a move...(in this case the only legal move!) and that's not allowed -- this is purely academic IMO, I don't think anyone is actually going to accuse anyone of cheating for pointing out the e.p. rule -- but I think if we were to be strictly rule conscious, we would have to say:
"There is no evidence here of a problem with the chess program/website, wait till the game is over (via draw by agreement, timeout, or a move is found) and we'll explain."
And then there's the "although my client in no way admits wrong-doing of any kind we would be willing to consider a plea-deal in order to..." answer:
"I can't tell you what move is available, but I can tell you the solution you seek is in the rules of chess which you are free to read. I'd start reading soon, beginning at the section marked, "en passant."

Not only is it not mate for the reason listed above but Black after the en passant capture has both of White's rooks in jeopardy and is a heavy favorite to win the game. Shame after such an (almost) neat mate with a pawn.

..."I can't tell you what move is available, but I can tell you the solution you seek is in the rules of chess which you are free to read. I'd start reading soon, beginning at the section marked, "en passant."...
Yeah, that's where I was trying to go. White can't claim a win. Black won't resign. What do you do? I like the option of saying there is a legal move but not telling what it is.
Apologies for the shoddy image, but I was playing this game on another website with a chess.com plugin. I'm white and just moved f4 (is that the correct annotation?) but the computer isn't recognising this as mate. Any ideas?