Start by examining every check you can give. There are two: Qh7 and Qh8. As you examine these, you also should notice that White’s knight can strike against the Black queen in one move—Nxe6; if Black’s king could be forced onto a dark square on which it could be forked at the same time, White would have the game. Does Qh8+ accomplish this? No; Black replies BxQ, and anyway if the king were to move to h8 it would be too far away from e6 for White to fork with his knight. What about Qh7+? The difference is that the queen would be guarded by White’s knight, which also attacks f7. Thus Black would have to move his king to f8—and into a fork. White plays Nxe6+ and wins the queen.

We aren’t finished. Black of course must move his king once it is forked, so press farther and ask where it will go. Moving it backward along the back rank is out, because White’s queen is there. He has to play Kf7. Does White then have any more checks? Yes; among others he has QxB—which is mate. The moral of this part of the tale is that when you have two pieces in the vicinity of the enemy king, and especially when one of them is a knight, always be mindful of the relationship between the pieces—how one can protect the other in an attack, how the knight can seal off escape squares in different directions by virtue of the odd shape of its moves, and whether the ultimate goal—checkmate—thus might be achieved.