You want to cultivate a constant awareness of the enemy king's status (not to mention the status of your own): constraints on its movement, pieces aligned with it that might be pinned, and any pieces of yours that are trained on its position. There can be great power in having two pieces—preferably a queen plus another—aimed at a square next to the king even if they can't both get there.
In the frame to the left, for example, White’s queen and a5 bishop both are aimed at d8. This is a mating threat. It doesn’t work now because Black has a pawn in the way on d6. But see the implication of this: the d6 pawn is pinned, since mate results if it moves off of its file. The pawn therefore is not doing any defensive work and anything it seems to protect is vulnerable—here, the knight on c5 and the pawn on e5. White is in position to take the e-pawn with Nxe5. (Following through, he asks about the consequences of putting the knight there. It will attack Black’s queen. Where will the queen go? Remarkably enough, it has no escape; every square to which it can move is under attack. White will take it on his next move.)
The trick to the position comes at the outset: you can't let the pawn on d6 prevent you from seeing the pressure White’s queen puts on d8. The pawn is not (or not just) an obstacle that ruins the mating idea for White. It is an opportunity, because in order to stop the mate threat the pawn must stay where it is, leaving its protectorates exposed to capture.