What is the black stuff in a digital watch display?

And how does it appear and reappear? Is it a substance? If so where does it go? Or some sort of optical illusion?

If I understand you correctly you’re asking “how do LCDs work”. Good explanation is found here.

Get a pair of polaroid sunglasses. Look at your LCD watch through them and rotate the watch. At some angle, the entire display will turn black.

The simple answer is that LCD changes polarization due when electrified.

      • Hmmmm…
  • How come the colors reverse when you flip the front polarizing film? That is, if you disassemble the whatever-it-is, there is always a plastic film on top of the glass LCD panel. If you flip that film over and then reassemble the device, the panel will then display white numbers on a black background, instead of black on white. I have seen this done several times but only with “black and white” displays, so I dunno what happens with a color panel… - MC

Read the link posted by therealblaze. It is very instructive.
Basically the liquid crystal rotates the direction of polarization, by virtue of having different n for the different components of the light.
In the ‘normal’ arrangement of the display the light goes straight through the ‘white’ area, but in the ‘black’ area the light gets rtotated to the polarization blocked by the front polaroid. If you rotate the polaroid it will instead let through the light that should have been absorbed, and vice versa.
As far as colours are concerned it gets a lot more complicated. I have only dissasembled one colour LCD and that was an experimental unit with three light sources behind (every pixel was subdivided into three, R, G and B) . If you were to rotate the front polaroid there you would simply get the corresponding vectors inversed. instead of [R,G,B] =
100%,100%,100% you’d get 0%,0%,0%,
[0%,0%,1000%] (blue) would turn into 100%100%,0%
(ie complementary colour, with inversed intensity?)

In fact the link posted by therealblaze is so good, that I’ll provide it again:How Liquid Crystal Displays (LCDs) Work.

The light reflecting off the face of the watch is already polarized in one direction. When you pass an electric current through the liquid crystal segment, the crystals line up in the direction of the current and polarize the light in the other direction so that light can’t pass though, thus the digits appear black. When the power is disconnected, the crystals return to their random orientations and the black digit disappears.

If you have some polarized lenses from an old pair of sunglasses, you can do a nifty experiment by placing the two lenses over each other and rotating one of them 90 degrees.