Light waves have a magnetic component and an electric component, as all electromagnetic waves do. These are mutually perpendicular, and extend perpendicular to the line of travel. Generally, light will be composed of randomly oriented waves. However, when it is passed through a polariser, the polarisations will all be resolved into one direction.
When the polarised light meets the second polariser (the analyser), it will again be resolved into that direction of polarisation. However, since the analyser is set at 90° to the first polariser, it should let nothing through whatsoever. With no sample between the polariser and analyser, no light is transmitted.
Glass, being isotropic, will have no effect on the polarisation of the light passing through it. Quartz has a crystal with a trigonal unit cell, and as such, it has unique properties along its c-axis. (a = b ¹ c, α = β = 90º, γ = 120º)
The refractive index for light travelling along the 'a' or 'b' axes (no) is 1.544; that for light travelling along the 'c' axis (ne) is 1.553. Light travelling through the crystal along any direction except  will encounter two refractive indices. It therefore splits into two parts: one part is resolved onto an ordinary direction; the other is resolved onto an extraordinary direction. These parts will travel at different speeds due to their different refractive indices. When the light emerges from the crystal, the overall wave is different from that which went in, and there is now some component of the wave parallel to the analyser. Some light may now pass through the analyser, as observed.
There is a special case of this, in which the light travels along . In this situation, only one refractive index is encountered (ne), so the light all goes through at the same speed with its polarisation unchanged. No light would therefore be transmitted through the analyser.
Even in the case of light travelling along a direction not parallel to , the quartz may be rotated so that the light emerging from it is exactly perpendicular to the analyser's polarisation direction. No light will be transmitted, as before. This explains why the intensity of transmitted light varied from zero to some maximum when the quartz was rotated. The minima appear at 90º intervals, being the angle between the c-axis and the other axis. This is observed in the animation above.