The excess payment offer could come for rent, as it often does, or for a chair, a table or whatever. The bad guys want you to cash their check and then "refund" them the difference because they only have one check or they had the cashiers check made out for too much or any other excuse.
Often, scammers will claim they are moving to the U.S. and need the excess to pay a mover who will only accept a wire payment from a U.S. bank, and the potential landlord will be asked to wire the difference. Well, as you might have guessed, their check will be worthless. The person complying with the request, whether selling goods or renting property, will be out that money. Some of the victims even have asked their banks whether the bad check had cleared. Unfortunately, it can take weeks for the fraud to be detected and the victim can still be on the losing end -- with their property vacant and their bank account depleted.
To learn more about overpayment scams, read this alert from the Federal Trade Commission.