Hydrogen peroxide diffuses through the dead, upper layer of your skin (epidermis) quickly and I'd guess it had already started doing this before you washed it off. It then moves into the lower, living tissue (dermis) where every cell contains an enzyme called catalase. The peroxide reacts with the catalase forming water and gaseous oxygen. The oxygen is released as microbubbles which cause microembolism of the capillaries in the areas along with the surrounding tissue. This causes the skin whiteness and the oxygen tries to rise out of the skin into the atmosphere. When complete, you can usually brush off the damaged skin a few hours later.
All cells produce hydrogen peroxide as a by-product of respiration. Catalase and another enzyme called glutathione peroxidase stops oxidative damage in the body which would be very harmful.