This is because of osmosis. What happens is that water will move into your cells. That makes sense of how skin swells up but it doesnít necessarily make sense that your skinís wrinkly. You'll notice when you have a bath you donít get wrinkles all over. You get wrinkles on the soles of your feet, on your hands: especially your fingers.
This is because you have a layer of protective keratin. Keratinís a kind of tough stuff, itís what claws and nails and so on are made of. This means that the skin here is much thicker than elsewhere on your body, and this thick layer is made of dead, keratinised cells. This takes up more water than the thinner skin elsewhere, and so swells up, and goes wrinkly.