Say you loan a few bucks to a friend who is short on cash, and he wins the lottery. Presumably, he'll be able to recompense you, or pay you back what he owes.
If you look closely at recompense, you'll see how similar it is to compensate, which means to make amends for something or to pay. Add the prefix re-, which suggests returning or repeating, and you get the meaning of recompense — paying back or reimbursing financially or otherwise. Recompense is most often used as a verb, but it works as a noun, too. In recompense for throwing peas and screaming, a mother might send a toddler to her room for a time-out.