As you may have worked out for yourselves, the winning line for white starts 1.Kd6, after which a typical line runs 1...Kf8 2.Ke6 Kg7 3.Ke7 (gaining the opposition) Kg6 4.Kf8 Kf5 5.Kf7, and white forces the pawn home.
This is not what happened in the game.
As Simon suggested in the comments, a plausible line that only draws is 1.Kf5?! Kg7 2.g6?? Kh6!, and then 3.Kf6 is stalemate, whereas everything else drops the pawn and leads to a drawn pawn ending.
Or at least, it should do.
What actually happened was that the game did indeed continue 1.Kf5?! Kg7 2.g6?? Kh6!, and white sat there for a while thinking about how she'd thrown a winning position away.
But she sat there thinking this on black's clock time. He'd forgotten to press his clock, and his flag eventually fell, giving white the win.
What this goes to show, I'm not quite sure.