Dermatologist Laurence Meyer of the University of Utah offers this explanation:

The epidermis, or outer layer of the skin, is made up of cells called keratinocytes, which form a very strong intracellular skeleton made up of a protein called keratin. These cells divide rapidly at the bottom of epidermis, pushing the higher cells upward. After migrating about halfway from the bottom of this layer to the top, the cells undergo a programmed death. The nucleus involutes, leaving alternating layers of the cell membrane, made of lipids, and the inside, made largely of water-loving keratin. The outer layer of the epidermis, called the stratum corneum, is thus composed of these alternating bands.

When hands are soaked in water, the keratin absorbs it and swells. The inside of the fingers, however, does not swell. As a result, there is relatively too much stratum corneum and it wrinkles, just like a gathered skirt. This bunching up occurs on fingers and toes because the epidermis is much thicker on the hands and feet than elsewhere on the body. (The hair and nails, which contain different types of keratin, also absorb some water. This is why the nails get softer after bathing or doing the dishes.)

Soaking in the tub does hydrate the skin, but only briefly. All the added water quickly evaporates, leaving the skin dryer than before. The oils that hold the water in have usually been stripped out by the bathespecially if soap and hot water are involved. But if oil is added before the skin dries, much of the absorbed water is retained. Thus, applying a bath oil or heavy lotion directly after a bath or shower is a good method of hydrating the skin.