Why do they call them flea markets?

The first theory, and the one most popular on the Internet, proposes that the term is a direct translation of the French Marché aux Puces, a large, outdoor bazaar in Paris. This original market earned its name from the critter-infested goods it was rumored to sell.

The second theory alleges that the term was coined at a time when the slums and alleys of Paris were demolished and replaced by new construction. The dealers in second-hand goods who lived and worked in these old neighborhoods were forced to flee. The merchants' new gathering place was referred to as the "flee market," which later became "flea market."

The final theory associates the term with New York City's 18th century Fly Market. Apparently, the Dutch name for the market was vlie, which means valley but is pronounced "flea."

http://ididnotknowthatyesterday.blogspot.com/2006/03/where-did-term-flea-market-come-from.html

The "flee market" suggestion from that site is another of those bits of fake etymology. The flea/flee pun only works in English and would never have been used in Paris, and the English would not have been involved with such markets at that time. They were selling junk not antiques.

https://en.wikipedia.org/wiki/Flea_market
lacks citations, excepting the one quoted by robz, and another which is an advert and probably shouldn't be there.