Then if C is aligned with A or B the light will still be completely blocked.
A and B have their axes at 90 degrees to each other... If A passes vertical polarization then that vertical vector has no component in the horizontal direction, so nothing is transmitted through horizontal polarizer B.
But... if you add C between A and B and if C is at 45degree angle to A and B, then vertical polarization passes through A, this vertical polarization DOES have a component along the 45 degree axis- so C passes that component. But now, the light passed by C is now polarized at a 45 degree angle and so it now has a component in the horizontal direction and so some portion of the light DOES go through B.

NOW, take just polarizers A and B, cross them so they block all light. Now put a piece of scotch tape between A and B- you will see a multicolored pattern of light passing through- scotch tape is birefringent, it rotates incident polarization of light, and the amount it rotates it by can be a function of wavelength.... so you see a pattern of colored light where the tape is.

Finally Take one of the polarizers, get a white piece of paper and look at it through the polarizer in bright light for a moment and then slowly rotate the polarizer.... a certain percentage pf people will see a pattern rotate with the polarizer- it turns out that a fair number of people can detect polarization with their eye- the effect is called Haidinger's Brush - is it a quirk of evolution, or was there a purpose for having that ability?

In the Lunar Laser Communication Demonstration the space terminal transmitted circular polarization- lets say it was left hand circular polarization. The ground terminal had a quarter wave plate that turned left hand circular polarization into Horizontal polarization.... we then used a horizontal polarizer which transmitted all of the horizontally polarized signal, but blocked ~half of the (mostly) randomly polarized sky background- thus improving our signal to noise ratio.

You can use polarizers to see stress in glass because stress can cause birefringence...