Black plays for black. The black knight blocks the only position that the black king can move to for safety, all other moves by the black king (or not moving the king at all) result in the king being captured. If it moves right it gets captured by the rook, move down or down-right, it gets captured by the white king. Don't move and it gets captured by the rook, and there's no way to play the black knight to block the rook, either.

Of course you could argue that if you're ignoring today's rules of the game that you may as well have the black king take the black knight, but that's illegal under the rules and probably has been right from the start.

The black knight plays for black.

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...