The basic technology behind the Indiglo watch is called electroluminescence. Electroluminescence is the conversion of electricity directly into light. This is not how an incandescent bulb works. In an incandescent bulb the electricity produces heat and the heat produces light. Electroluminescence is much more efficient because it converts the electricity directly to light.
The most common example of electroluninescence that we see on a regular basis is a neon light. In a neon light, high voltage energizes the electrons in neon atoms and, when the electrons de-energize themselves, they emit photons.
In an Indiglo watch a very thin panel uses high voltage to energize phosphor atoms that produce light. The panel itself is extremely simple. As described in the Timex patent (see the first link below), you take a thin glass or plastic layer, coat it with a clear conductor, coat that with a very thin layer of phosphor, coat the phosphor with a thin plastic and then add another electrode. Essentially what you have is two conductors (a capacitor) with phosphor in between. When you apply 100 to 200 volts AC to the conductors, the phosphor energizes and begins emitting photons.
Creating the high voltage can be a problem in a wristwatch. The watch has only a small 1.5 volt battery. To produce the 100-200 volts, the 1:100 transformer is used. By charging the primary coil of the transformer with a transistor that is switching on and off, the secondary rises to 150 volts or so.
Here are several interesting links: