If you move the Queen from H3 to H4, it attacks the king. The king can't move to E7 or G7 (because there are black pieces), can't move to E6 or F7 (bishop attacks those tiles), can't take the knight on G6 (defended by a pawn), can't move to E5 (attacked by the knight), can't take the pawn on F5 (defended by another pawn), can't move to G5 (also under the attack of the Queen). The king is under attack, can't move in any of the 8 directions and nobody can take or block the Queen in H4.
Now, I haven't double checked if there is another solution in quite some time, but the one proposed in the walkthrough is absolutely correct.
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I think missed the knight, so you're probably right about that. now that I look at it, the black rook ends up blocking the bishop move anyway. I guess I really am bad at chess for a reason.
Well, I found a minor visual oversight at least; there's still 3 bishops. It's probably safe to swap the leftmost white-on-black bishop to a rook.