To befoul something is to make it really dirty. If you befoul your shoes in the mud, don't wear them inside the house or you'll also befoul the carpet.
You can use this verb to mean pollute or contaminate: "Dumping those chemicals down the drain will eventually befoul our town's drinking water." It's also good for anything that gets gross and dirty, like when you befoul a brand new sweater by spilling mustard on it. Figuratively, befoul means "damage," the way gossip can befoul someone's reputation.