Why can two clear-looking polarizing sheets become dark together?

Two polarizing sheets can look fairly transparent separately yet appear almost black when placed together in a particular orientation. Rotating one sheet can make the pair bright again. The important change is the relationship between their transmission axes, not the amount of material in the light's path.

Light is an electromagnetic wave. For a beam traveling in one direction, its electric field oscillates across that direction. In unpolarized light there is no single fixed oscillation orientation. An ideal linear polarizer transmits one selected component, leaving linearly polarized light with a definite axis.

The second sheet selects the component along its own transmission axis. If its axis matches the first, the ideal second sheet passes the already polarized light. If the axes are perpendicular, none of that polarization is aligned with the second axis, so the ideal output is zero. The axes are directions, not tiny open slots drawn to scale.

In the ideal intensity account, the first sheet passes half the incoming unpolarized intensity. With a 45-degree angle between axes, the second passes half of that remainder, leaving one quarter of the original. Real sheets absorb some wanted light and leak some unwanted light, so actual transmission is not perfectly one half or perfectly zero.