While wearing polarized sunglasses, it is sometimes impossible to read the LCD display on calculators or electronic wristwatches. The display will be completely black. Why does this happen?

The structure of your LCD display (in a calculator or watch) is like a sandwich. You have a polarizer (Pol.1) a sheet of liquid crystal and a second polarizer (Pol.2).

The two polarisers are crossed so no light passes through, but the liquid crystal has the property of "twisting" light (rotate the Electric field; have a look to "elliptically polarized light") so that through Pol. 2 passes light (your display looks grey not black).
When you "activate" the liquid crystal (through one of the electric connections) you change the properties of it (at a particular position) so that now it doesn't twist light anymore. The light (horizontally polarized, for example) passes through without changes and gets blocked by crossed polarizer, Pol. 2.

When you look at your display (showing #1# for example) you see grey (light the was twisted and passes vertically polarized through Pol. 2) and the number, formed by "blocked" light.

If you wear polarized glasses it is possible that the axis of polarization of them is crossed with respect to the one transmitted by the lcd (around the number #1#). So you'll see black for number #1# and black for the display, i.e., all black!