Explanation

A pinned piece cannot move at all (or only in the direction of the attack) without losing material. It cannot run away, and therefore defending against another attack on the pinned piece might not be possible.

Examples

Black's bishop e4 is pinned by the white rook on e1 against Black's king on e8.
White can attack the pinned piece with Bf5, and Black has no way to defend it.
Note that Bf3 would not have been the correct move, as Black then would be able to defend the bishop with f5.

Black is a bishop down in this position, but can regain the material with an attack on a pinned piece.
White's bishop g5 is pinned by the black queen against White's unprotected queen.
Black can attack the pinned piece with h6, and White has no way to defend the bishop.
Note that f6 would not have been the correct move, as White would then be able to defend with an intermediate check Qe6+ (or Qc4+).

Black's knight f6 is pinned by the white bishop on g5 against Black's queen on e7.
Black has just attacked the bishop with h6, but White can ignore that and attack the pinned piece with e5.
If Black captures the bishop with hxg5, White can capture the knight with exf6.
This opens up the e-file, and White is winning because of another pin: Black's queen on e7 is pinned by White's rook on e1 against the king. Be6 doesn't help, because the queen is now also attacked by White's pawn on f6.

Related motifs

If the pinned piece is protecting another piece, that apparent protection can turn out to be false, and the pin can be used to win material.