Why do your fingers wrinkle in the Bath?

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 doesn't necessarily make sense that your skin's wrinkly. You'll notice when you have a bath you don't 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. Keratin's a kind of tough stuff, it's 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.

----UPDATE-----

In 2013 UK scientists discovered that fingers go wrinkly in water to improve our grip on wet or submerged object. They also showed that it is in fact constricting blood vessels, not osmosis, that drives this change.

http://www.karger.com/Article/FullText/328223