The outermost cells of the skin die and slough off because

The keratinocytes: cells found in the outermost layer of the epidermis that produces fibrous protein which gives hair, nails, and skin their hardness and water-resistant properties dies as a result of their distance from the rich blood supply the cells of the next layer lie on top off. Due to less nutrients and oxygen, these cells die as they are pushed towards the surface of our skin.