We see a cloth's colour by the light which hits it. For example, when white light (containing all colours) hits a green shirt, all the colours in the light are absorbed by the shirt,except green, which gets emitted. This emitted green light is why the shirt looks green to our eyes.
When the shirt becomes wet, water is absorbed in the fabric, including the outer layer of the material. When white light hits the surface of the material where it is wet, some of the light will scatter (refract) from the water. This light doesn't make it back to our eye. Less light coming from the wet region means it will look darker. Not all of the light that is emitted by the green fabricreaches your eyes now. Because there is less light coming from the wet area (some was refracted in a different direction), that area will look darker than the material around it.