In the solution presented at the end of the article, whose turn to move is it? Who can't make any further moves coz they've been check-mated? That's what I don't get. If it's white, white can still move their king back towards their home row and there's more than one move left in the game. If it's black, black can move the black knight in between the two kings and we'd still have more than one move left.

White advanced the pawn and promoted it. It is black's turn. Moving the knight between the kings is illegal because the black king is in check by the white rook.

Black is in check from the white rook. Moving the black night doesn't escape the check, so isn't a legal move.

Now I get it. The "beef" is that white replaced their white pawn with their opponents black knight, which of course the black king can't eat...