Why is moving King to Kf8 here much better than Kd8?

@lesnar, can I still castle if I have moved my king to avoid this check?

Nope because you moved the king

Well, this thread is definitely past the point where providing an answer would help the OP, but since someone else already necroposted it can't hurt too much.

(Side note...how in the world did Black end up with two pawns on both the f-file and the g-file?)

I think the reason Kf8 is better is because with this move, Black moves the king away from White's army, rather than towards it. White must now reroute if they want to attack the king again, which is not easy and would take a lot of time. Meanwhile, Black can play Re8, and from there the position will likely simplify into an endgame where Black is noticeably better.

Castling is overrated.

Kf8 moves the king away from all of white's attacking pieces....see that great tsunami of white pieces in the queenside...the king wants to move away, not towards that tsunami...
Hope it helps!

Hmm. After 11 O-O-O, Black was a pawn up and had an advantage according to Stockfish.

It's rather pointless digging at a 4,5 y.o. thread, but anyway: 12...Kf8 is much better than 12...Kd8 because artificial castling and connecting the rooks is just 2 moves away (...g6, ...Kg7).