The evaporation of water drains heat from the body of the animal (this is a common cause of hypothermia), requiring the animal to burn calories to maintain its body heat, calories better spent hunting, mating, keeping your immune system ( or caring for other members of the community (family, flock, herd...).
Also, the evaporating water might make small particles from the animal's fur, feathers or skin airborne, thus spreading the scent of the animal over a larger area than usual, attracting predators.
Yet another reason is the disabling effect the water might have on the fur or feathers of the animal. In cold weather, an ice crust might form on the wet fur and the moist might accumulate snow and grime, and the extra weight of the water might make flight impossible.

Trust me on this; I've got fur.