This is a very good question; the reason it's taken me so long to answer is that I've been thinking about it for quite some time and discussing it with friends. I don't have a definite answer; all I can offer is a hypothesis.
Some observations: First, it's not just water that does this. A drop of oil on paper or clothing produces a similar effect. Second, it doesn't always make things darker. A wet spot on cloth or paper looks dark when you're on the same side of the cloth as the light source, but it looks lighter than the rest of the cloth when you look through the cloth toward the light. So liquids like water or oil seem to increase the transparency of porous materials like paper or cloth: they get easier to see through rather than actually darker.
One thing paper, cloth, soil, concrete, hair, etc. all have in common is that on a microscopic scale, they're partly transparent. Rock, sand and soil have lots of tiny, colored crystals in them. The fibers in hair, paper, and cloth look like transparent glass rods close-up. Most light-colored substances are actually composed of small clear bits: they look white because the light reflects off the surface of the clear bits, bouncing around like a billiard ball before scattering in all directions.
Things which aren't white, like colored sand, concrete, or hair, are often made of clear but colored substances, like a brown or green beer bottle. Notice that some light reflects off the surface of a beer bottle, but the light that gets through the surface gets colored because the glass absorbs some colors of light.
Light will reflect off a surface for one of two reasons: either the surface is a conducting metal, or the substance has a different "index of refraction". When light hits a conducting metal, it's as if it hit a brick wall: it has no choice but to turn around and go back. You can think of light travelling through materials with different indices of refraction as being like a crowd of people walking over sections of floor at different heights. When the crowd comes to the high "step" where a raised piece of floor begins, some of them cand climb up the step and keep going, while others decide it's not worth the effort and turn around to go back. Similarly, when light hits a material with a different index of refraction, some of it is reflected, some of it is transmitted.
Most solids, like rock, glass, hair, cellulose (cloth and paper), and plastic have a very high index of refraction. Air has a low index of refraction, so when light travelling through air hits a solid, lots of it is reflected (it's a very high "step up"). Liquids like water or oil have an index of refraction between air and most solids, so if you put a layer of water over a solid like glass or paper, you put a "halfway step" between the two "levels of floor". Just like a stairstep, this makes it much easier to go from one "level" to the other: more light goes through into the solid than when there was no water.
So you see that by providing an index of refraction between air and a solid substance, you reduce the amount of light reflected by the substance. If less light is reflected back into your eyes, the substance will appear darker. If the substance is between you and the light, more light will be transmitted through the substance to your eyes.
You can probably test this hypothesis. Look up the indices of refraction of various materials like glass, polyester, and cellulose. Look up the indices of refraction of various liquids like water, oil, alcohol, or glycerine. Wet the different solids with the different liquids, and see if the darkness of the wet spot changes with different liquids. Look in an optics textbook to find out how much reflection should occcur at the interface of objects with different indices of refraction, and see if your results make sense.
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