When you pin a piece to the enemy king, the king not infrequently may protect the target against capture. In that case you sometimes can loosen the target by throwing a check at the king to drive it away from guard duty. The cue to look for, and to notice in each of the next positions, is a piece not only pinned to its king but on a square adjacent to it.
In this first example White again starts with a pin, this time of Black’s knight on f6. Count the attackers and defenders of the pinned piece. There are two of each, so the knight seems safe; but it isn’t. The pinned piece is adjacent to its king and receives important protection from it—a type of protection that tends to be unstable. To exploit the situation White just needs to get a piece within harassment range of Black's king. His rook would be ideal, but its path to the action is blocked by White’s own knight on the e-file. We've seen a way to deal with this, though: clear the White knight out of the way by letting it take the pinned piece, as with NxN. This is a powerful move, since it not only takes a piece but threatens the fork NxB—which also would be a discovered check against Black’s king. So Black must take the knight, and the only piece he can use for the purpose is his queen. Once Black plays QxN, his queen has interposed its way into a pin. It’s a shaky pin, as the Black queen is guarded and can take White’s queen with impunity if given a chance to do so. But White doesn’t give him the chance. Having cleared a path for his rook, White brings it to bear—not on the pinned piece, but on the piece protecting it (i.e., the king). Re7+ requires Black to move his king away from the job of guarding his queen. (QxR obviously is out of the question for Black; remember that a piece pinned to its king can do no defensive work.) Black plays his king to the back rank, and White has QxQ on the next move.
The point: when a pinned piece is protected by its king, a check may suffice both to remove its protection and to keep your opponent busy answering your moves. It’s worth some labor—i.e., making some threats and exchanges—to create avenues for such a check.