Capturing the pinned (or potentially pinned) piece, and so causing it to be exchanged when your opponent recaptures, can have pleasing side effects apart from an upgraded target. It may serve to loosen the square you need or to open lines you can use to rally offensive firepower against the pinned position.
Here's an easy one. The first thing to spot in this first illustration is the Black knight in line with its king—a kernel of a pin that must not be overlooked. Next look for a piece with the power to impose the pin; see that the bishop on b3 can get to d1 in one move. Now size up the impediments to the idea. The pinning square, d1, is guarded by the rook on d2, which also protects the knight. What to do? One way to move a bothersome enemy piece is by taking something it protects. In this case the bothersome piece is the Black rook, and it protects the knight on e2 (as well as the pinning square). So White plays RxN, Black replies RxR, and now the pinning square has been loosened—and the target of the pin has been changed to a rook. Bd1 paralyzes the rook and wins it a move later, as Black has no resources available for its defense.