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My opponent said it was the "worst game of his life"

White messed up bad by playing 6 g3??

The move 6 Ke2! is actually kinda solid. I mean ya the king is walking into the center, but at the same time only the black queen is out, and the black light square bishop is at least 3 moves from coming into play. Get that knight to f3, and the king over to f2, and white is up a piece smooth sailing.

After move 6, white is toast. Black missed some much more decisive moves like taking the hanging f2 knight on move 10, but the moves were all good enough to keep a recovery from being possible although it could have been done quicker good job.

3 f3 was bad. 3...e5 would have punished it better than 3...e6. Look at the game and put the pawn one square further: you control d4, Bc8 can come out...
Computers like 6 Ke2, but in a real game between humans 2 pawns for the knight + exposed King often is enough compensation. It is not necessary to find a mate: winning a 3rd pawn often allows to win the endgame.

3. f3??
As grandmaster Ben Finegold keeps saying, never play f3.

Anyway, neither of you played too well. But you punished his mistakes sufficiently, and that's all that matters.