No one is certain why the police are called the fuzz. According to The Straight Dope, what linguists know for certain is that "fuzz" was first used to refer to policemen by criminals in the United States during the late 1920s and early 1930s.
According to World Wide Words, this use of fuzz may be a variant of the word "fuss." This could be because dealing with the police causes a fuss or that citizens sometimes find officers to be fussy. World Wide Words also suggests that fuzz may be a mispronunciation of "Feds," meaning federal agents. This explanation is plausible since the use of fuzz originated during Prohibition in the United States, 1920-1933, when federal agents worked with local police to catch moonshiners and speakeasy operators.
Evan Morris, known as the Word Detective, points to references in glossaries of criminal slang compiled at the time as proof that the term really was in use in the criminal underworld. The book "American Tramp and Underworld Slang," published in 1931, is one of the best examples. According to Morris, the most likely explanation is that criminals used fuzz as a specific kind of insult. Use of the term “fuzzy” to mean “unmanly” is common in criminal slang of the day.