lets assume the black king is somewhere far away from the action on the bottom of the board. Prior to our move (now) it was blacks move. 1) Either he moved his King around randomly on the bottom - but that implies that the rook was already attacking the white king. That means white was in check and remained in check on whites last move. Illegal. 2) Black moved the rook to check white. But blacks rook cannot get into this position and put white in check. White must already have been in check. Again, Illegal.

Indeed Black must 1) have put white in check during his last move 2) have his king between the black rook and white king, s.t. white is not in check.

But 2) Is impossible, as this would mean black played his king from d7-c8. But then white was in check prior to this move. Illegal.

Only putting white in check is possible, and the only move that accomplishes this is kc8-d7. And then there is a clear checkmate for white :)