Figure[White to move]

You see that Black’s bishop attacks your rook. Before moving anything, though, think about offense—e.g., captures you can make. Your rook attacks Black’s rook, which is guarded only by its king. Mission: drive Black’s king away so the rook on d8 is left loose. What tools do you have for this purpose? For openers you have a knight that can give check. Think about Nf6+. If Black moves his king he loses his rook, so naturally he instead would prefer to play e7xN. This appears to ruin your idea, but not so fast: that check and capture would have had significant consequences, removing the Black king’s pawn cover on the e-file. Remember the mission, and consider checks you then would have (that wouldn't require moving the d5 rook from its file). There is one left: Rh1-e1+. This time Black has no choice but to move his king and forfeit his rook. White wins the exchange; since he operated solely with checks, Black never was able to play BxR.

Again we see that it may take multiple checks—whether with the same piece or successive pieces—to force the king away from its protectorate. Don't give up on such a possibility too easily.

For an additional little illustration of the same principle, imagine this same position with the g2 pawn off the board. Then after 1. Nf6+, e7xN; 2. Re1+, Kf8, White has 3. RxR+, Kg7—and then Re1-g1+, forcing the Black king away from the h8 rook and allowing it, too, to be taken.