Figure[Black to move]

You might begin by observing the battery of queen and bishop Black has aimed at f1, next to White’s king; he would mate there were it not for the knight on e3 that protects the needed square. The other thing to see is that White’s king is trapped on the back rank, nearly able to be mated with Rd1—another possibility thwarted only by White’s knight on e3. Thus while the e3 knight protects no pieces, it protects two mating squares, and this is one too many. Rd1+ makes White’s position untenable. He has to take the rook with his knight, or White is mated now (interposing the other knight on e1 just delays things a move). But if he does play NxR, Black mates with his queen on f1.

An initial lesson here is to look not just for the first mating threat you see, but for all of them; for if there is more than one, it may be that attempting to execute one of them (and failing) allows you then to carry off the other successfully. Another point to take away: when you see an enemy piece guarding a square you need for mating purposes, ask not just what pieces it might also guard but also what other sensitive squares near the king it protects.