The skin is covered with its own special oil called sebum, found on the outermost layer of skin. Sebum moistens, or lubricates and protects oour skin. It also makes the skin a bit waterproof. But staying in water for a long time washes away the sebum. Then, the water can penetrate the outer layer of your skin. This causes your skin to become waterlogged. Some scientists believe that this leads to wrinkles since the skin expands to allow extra water inside. The expanded skin ends up looking really wrinkly. Others think that it's because the skin is tied down to the tissue underneath in certain places. So when the skin is full of water, it swells up, but only in places where it is not tied down, which makes it look wrinkly.
Usually the tips of the fingers and toes are the first to wrinkle because of a thicker layer of keratin and an absence of hairs.
(Water probably initiates the wrinkling process by altering the balance of electrolytes in the skin as it diffuses into the hands and soles via their many sweat ducts.)