In the positions just seen, the attacking player used his rook to give check. The opponent’s need to defend against the check gave the piece that had been uncovered by the rook time to win material on the next move. Sometimes the rook can achieve much the same effect by threatening mate, as this puts the enemy under pressure similar to that created by a check. A rook most commonly can threaten mate by offering to drop onto the back rank when the enemy king is trapped there. This theme is worth our independent attention for two reasons. First, seeing a mate threat often is harder than seeing a check; it takes some practice. Second, when you work with mate threats you have to be especially careful to consider what trouble your opponent can make in reply. Since by assumption you haven't put him in check, you may have left him with latitude to respond with checks of his own that seize the initiative.
In the current position White has the kernel of a discovered attack along the g1-a7 diagonal, where his rook masks his queen—which otherwise would be able to take Black’s queen. Now think about execution: what large threat can White create with the rook? Rxf6 is its only capture, but the move is more than that: it aims the rook at f8, where it would mate the trapped Black king with RxR#. So in reply to 1. Rxf6 Black has to capture White’s rook (g7xR), which of course creates time for White to play QxQ on his next move.
Now notice that if it were Black’s turn to move he would have Qa1+. White would be forced to reply KxQ—after which Black has the knight fork NxB, winning a piece. This should worry you a little for the reason sketched a couple of paragraphs above. The first move in White’s planned sequence (Rxf6) is a capture and mate threat, but not a check; so you have to consider whether he can give checks of his own that will derail your plans. In this case Black could respond to Rxf6 with two consecutive checks, as just discussed: Qa1+, and then NxB+, forcing White to move his king. So are White's plans foiled? No; for then Black is out of checks that hold the initiative: the next move he wants to play is NxQ, but he can’t afford it because Black mates with RxR. White thus holds onto the queen.