The answer lies in the structure of our skin, and that it's not the same all over. The outer layer of skin, the epidermis, on our palms and the soles of our feet is thicker than the skin on the rest of our body. When we soak in water the epidermis all over our body absorbs some water and swells. Because the skin on our palms and soles is thicker it can absorb more water, and therefore swells more.
That seems simple enough, but why do the ends of our fingers and toes wrinkle up like prunes and not the rest of us? Most of our skin absorbs water and swells rather evenly, but there is something different about the ends of our toes and fingers: there is a nail on one side. The ends of the fingers and toes can't swell toward the nails, so the skin bunches up opposite the nails on the palm side of our fingers, and the sole side of our toes. This bunching up of the skin gives us the wrinkles -- prune skin.