Question

Why does the skin of our finger shrink when we wash clothes

Answer 1

because concentration of water outside the skin is lower than our skin's concentration

Answer 2

Our skin of finger shrinks when we do the same, as when we wash the clothes the water around the fingers or hand behaves like hypertonic solution and water starts coming out from finger(cell) to out side region causing the fingers to shrink.
Because of exosmosis

hope it's helpful ☺