That grey piece of plastic is a polarizing filter. A polarizing filter passes only (more or less) light which oscillates at the same angle. As an example it oscillates verticaly, i.e top to bottom of the display. In a calculator you have usualy two of this filters with the same orientation. The liquid cristals are between them, like a sandwich. Normal light from a bulb or the sun oscillates in every possible direction. If light gets on the display only verticaly oscillating light can pass the first filter, the liquid cristals and the second filter. There it will be reflected and does the same way back. You will see a bright display.
Liquid cristals can change the direction of the light oscillation and that effect can be electricaly controled.
So with that effect turned on, light passes again the first filter and becomes verticaly polarized. Than the liquid cristals turns the polarization of the light by 90 degree and the light can not pass the second filter. You get a dark display.
Without the first filter the light doesn't get poliarized and it does not matter if that unpolarized light will be turned or not. A part of it will allways pass the second filter and find the way back. You get allways a bright display.
P.S. The text above describes not the only possible arrangement of filters and cristals, there are many more.
If you are interested have a look at LCD on the wikipedia