How does dye work?

Answer 1

It's not clear exactly what you're talking about.

If you're asking why dyes are colored, it depends on the dye, but for the most part it has to do with the electronic properties of the chromophore (the chemical in the dye that gives it its color); the details are complicated to figure out and certainly too complicated to explain here.

If you're asking how the dye colors other materials, again it depends on the dye. Some dyes are called mordant dyes, which means that they chemically "latch on" to the molecules of the substance they're coloring. Others just absorb to the surface of the substrate; these tend to be not as colorfast and the dye will "leach out" of the material over time. Finally, some materials have the dye physically trapped within the substance itself. This method is common for certain kinds of plastics, where the dye is added to the polymer resin before it's actually polymerized or to the molten polymer before it's formed into strands (or whatever). This allows the material to be colored all the way through and not just on the surface, so the color won't be affected much by surface wear.