Black’s threat is to capture the rook with the bishop, then trade the pawn on d2 for the other rook. The remaining two pawns cannot both be stopped by the white king.

White needs to create a threat urgent enough that black cannot waste time capturing the rook.

Black being in check is a red herring, as black can relieve the check by simply capturing the rook. White needs to threaten mate-in-1 with the other rook.

The king is on b6 with the threat of a back-rank checkmate. The bishop and knight are not helpful at defending, so black must play Kc8, allowing white to play Rxd2. This stops both of black’s threats while renewing the threat of back-rank checkmate. There is no defense without sacrificing a piece, after which white mates easily.

Were the king in any other position, black can take the rook and maintain a decisive advantage.

What move does black want to play to defend?

The white king must be positioned in such a way that black cannot castle.

For castling to be legal: you cannot castle from, through, or into check; neither the king nor rook can have moved; and the intervening squares must be unoccupied.

This is a retrograde chess problem.

Is there a location for the white king such that to get there the black king must have moved?

The white king is on g8, and white plays f8 checkmate regardless of black’s move.

If the white king is on g8, how did it get there? Four of black’s pawns have never moved, so the white king has never been to b6, c6, d6, e6, f6, or g6 and must have passed through either a6 or h6 on the way to the seventh rank. The move h6 to g7 is impossible because the pawn on h6 came from g7 and has occupied one or the other square the entire time; likewise for a6 to a7. Therefore the only way for the white king to have moved from the sixth rank to the seventh rank is a6 to b7. To get from b7 to g8 the white king passed through e8, so at some previous point in time the black king was not on g8, and therefore it is illegal for black to castle.

If the white king was on h8, black could defend with Be5. Qxe5 walks into Kf7 checkmate, and any other move lets black capture the white queen, quickly leading to black winning.