Water-induced wrinkles are not caused by the skin absorbing water and swelling. They are caused by the autonomic nervous system, which triggers localized vasoconstriction in response to wet skin, yielding a wrinkled appearance. This may have evolved because it gives ancestral primates a better grip in slippery, wet environments.

Evolutionary neurobiologist, Mark Changizi of 2AI labs, has a new hypothesis on why -- but not how -- this occurs. He analyzed the wrinkles in 28 human fingers, and concluded that the morphology of wrinkled fingers has similar properties to drainage networks flowing from mountains. He calls it the 'rain tread' hypothesis. Wet-induced finger and toe wrinkles, he said, act as drainage networks that channel away water to allow for a stronger grip when it's wet.

Upon continued submersion in water, the glabrous skin on human hands and feet forms wrinkles. The formation of these wrinkles is known to be an active process, controlled by the autonomic nervous system. Such an active control suggests that these wrinkles may have an important function, but this function has not been clear. In this study, we show that submerged objects are handled more quickly with wrinkled fingers than with unwrinkled fingers, whereas wrinkles make no difference to manipulating dry objects. These findings support the hypothesis that water-induced finger wrinkles improve handling submerged objects and suggest that they may be an adaptation for handling objects in wet conditions. Changizi's theory has not been proven. Aside from one picture of a Japanese macaque with pruning fingers, he has no direct evidence on how many other primates share this ability.

And this article in Nature News quotes Xi Chen, a biomechanical engineer at Columbia University, who calls the theory "unjustified":
Chen thinks that the wrinkles have a simpler cause: when fingers are immersed in hot water, the blood vessels tighten and the tissue shrinks relative to the overlying skin. This contraction causes the skin to buckle. "It's a classic mechanics problem," he says.