When you slough, you get rid of the rough. To slough is to remove an outer layer, like filing dry skin from feet. You can slough away emotions too, like the heebie-jeebies you get thinking about dead skin from people's feet. Ew.
Slough rhymes with "rough." It doesn't sound as though it'll give you a beautiful result, but when you slough away old skin, new skin appears. Snakes shed or slough off their skins as they grow and get rid of icky cells, and humans do the same, though fortunately we don't slough off one big skin as snakes do. Maybe it's best to slough off that mental image with a nicer one.