Assess the threats to your own king and you see that Black is threatening mate with Qxh2 (as well as the somewhat less formidable, but still troublesome, QxR+). It's hard to see a satisfactory way to avoid both threats by moving the White pieces around defensively. But we know the value of not reflexively playing defense; in this case, by taking the time to examine the checks he can give White finds he can disregard the threat and achieve mate himself. The first thing to see is that White almost has a back rank mate with Re8, but Black’s rook on f8 blocks the way; and then there is the coordination of White’s queen and bishop against f7. If Black’s rook could be drawn up to f7, it would become pinned and enable the back rank mate to work. You equally might see the idea by just looking at any checks you can give and their consequences. Either way the result is the same: Qxf7+ requires RxQ in response—drawing Black's rook into a pin. You take advantage of this by noting the defensive purpose the rook had served and exploiting the new vulnerability with Re8#.