Why do shields in Star Trek "wear out"?

When a ship is under attack, the shields wear down over the course of the attack.

What is the mechanism behind this? Aren't the shields powered using energy? While the ship has energy, shouldn't it be able to indefinitely defend against attacks the shield can absorb?

Answer

According to the Star Trek: The Next Generation Technical Manual (by Okuda / Sternbach), Section 11.8, pg. 138 ("Deflector Shields"):

Like most forcefield devices, the deflector system creates a localized zone of highly focused spatial distortion within which an energetic graviton field is maintained. The deflector field itself is emitted and shaped by a series of conformal transmission grids on the spacecraft exterior, resulting in a field that closely follows the form of the vehicle itself. This field is highly resistive to impact due to mechanical incursions ranging from relativistic subatomic particles to more massive objects at lesser relative velocities. When such an intrusion occurs, field energy is concentrated at the point of impact, creating an intense, localized spatial distortion.

This zone of spatial distortion around a ship is created by the so-often-mentioned shield generators:

The deflector system utilizes one or more graviton polarity source generators whose output is phase-synchronized through a series of subspace field distortion amplifiers.... Heat dissipation on each generator is provided by a pair of liquid helium coolant loops with a continuous-duty rating of 750,000 MJ. Four backup generators are located in each hull, providing up to twenty-four hours of service at 65% of nominal rated power.

It is clear then that several things may go wrong when an "intrusion" through the shields is attempted. First of all, the distortion of space around the ship (what we call "shields") is designed to adjust itself rapidly from a diffuse state to a localized one, to provide maximum resistive force at the target site of the intrusion. This is an enormous drain on energy supplies. Volleys lobbed at disparate points in a short time would cause the biggest drain of this kind.

Secondly, the shield generators produce considerable amounts of heat, which need to be dissipated. This heat output increases when the shields localize. If the generator coolant loop fails on a particular generator, that generator must be taken off-line until diagnostics and repairs can be carried out, further compromising the integrity of the shields.

In short, stressed deflector shields cannot function "indefinitely".

What does it really mean to suggest that the shields can continue to function "indefinitely" because they can absorb energy? There is no reasonable or efficient way to transfer this energy back into the shield generators. The shield generators are located at precise points along the ship. When the shields absorb energy, this energy is dissipated in various forms, e.g. mechanically (by disturbing the shape of the field generated by the shields and/or the position of the ship) and as light and heat. There is no way to direct this energy coherently back into the shield generators, or into any collection point along the hull, I'm afraid.

Note: The Technical Manual is (almost entirely) canon. Okuda and Sternbach were designers and technical consultants on TNG, and the introduction for the manual was written by Roddenberry himself.