E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
A broom is hung at the mast-head of ships about to be sold, to indicate that they are to be swept away. The idea is popularly taken from Admiral Tromp; but probably this allusion is more witty than true. The custom of hanging up something to attract notice seems very common. Thus an old piece of carpet from a window indicates household furniture for sale; a wisp of straw indicates oysters for sale; a bush means wine for sale; an old broom, ships to sell, etc. etc. (See PENNANT.)
A new broom. One fresh in office.
New brooms sweep clean. Those newly appointed to an office find fault and want to sweep away old customs.