opticsreflectionvisible-lightwater
It's a well-known fact that when one soaks a thin piece of fabric, it will often become more transparent than it was before.
What is the reason behind this? I can't put glass behind the fabric and increase its transparency.
(Also, just in case this seems like a duplicate, I am not concerned necessarily with why the fabric become darker but just the transparency aspect, unless, of course, the causes of these two phenomena are not disjoint.)
The thickness of the AR coating is chosen such that the reflections from the two interfaces cancel out (at the wavelength for which the AR coating was designed):
See Anti-reflective coating in Wikipedia.
As endolith points out in the comments, to explain how the transmission is enhanced, you have to draw a few more rays in the diagram. Here's another illustration, from the Wikipedia article for Fabry–Pérot interferometer, which shows a few higher-order reflections:
For the anti-reflective coating, you choose the thickness such that R1 and R2 cancel while T1 and T2 constructively interfere. Note that this is dependent on the wavelength, the angle of incidence, and the index of refraction of whatever is being coated. With other thicknesses, you can make a high-reflectivity coating, or a coating of whatever reflectivity you want.
A truly transparent object doesn't reflect any color in the sense that you seem to mean. How we normally "see" transparent objects is by the refraction of light that they cause and the effect on the image of what lies "behind" them. There is, however, a phenomenon called "total internal reflection" where a transparent object can reflect ALL light that approaches its boundary within a certain range of angles. This is what causes the strange reflection you see when looking up from under water near the surface and how fiber optic systems work.
Best Answer
The fabric is made of many thin fibers of fabric, with air in between. This structure causes light to bounce around many times inside, making it hard for light to get through.
When you make the fabric wet, you replace the air with water, which has a closer index of refraction to the fibers. So the reflections inside are less important, and more light just goes straight through.
Some people might interpret this as the clothing getting darker, especially if somebody is wearing the clothes, but that’s just because no light is coming from the other side. If you hold the fabric up to the sun, it should be clear the effect is really just an increase in transparency.
For the same reason, paper becomes transparent when it gets oily, as anybody who has ever eaten pizza on a paper plate knows.