[H8crilA]

Let's follow energy, after all darker = less energy. For simplicity let's consider the sum of energy over all observer angles, to get rid of the "it's shiny only from one angle" problem.

Reflection on the surface of the water is "free" both ways (from air to air bounce, and from water to water bounce), no energy loss. Travelling through water - similar, lite energy loss, though not exactly zero. The only way we can really lose energy is on the cloth reflection. Therefore (almost) all of the darkness effect should be attributable to increased number of cloth reflections, via the total internal water reflection mechanism.

This also explains why colors are more vivid when the cloth is wet - after all the same ray of white light bounces off the cloth several times, increasing the "color filtering properties" of the cloth.