So, from the other answers provided to this question, you can see that "fleshpots" has more than just a sexual meaning; in the context of the Bible, it referred literally to bowls of cooked meat, and implicitly to the physical comfort that was provided to the Israelites while they were slaves in Egypt (a loose translation of Exodus 16:3 might read "If only we had died in Egypt, at least there we had great dinners. Moses is going to kill us with hunger." In short, fleshpots represent temptation combined with a loss of morality, sort of a Faustian trade of one's soul for worldly possessions.

I assume that this word is appearing in a description of Antony & Cleopatra, since it isn't in the text itself. It seems most likely to describe Antony's relationship with Cleopatra, in that her seduction has led him to forget his responsibilities and become infatuated with the comfort she provides. This is further evidenced later in the play by Antony's repeated forgiveness of Cleopatra, such as when she "makes a coward of him" by fleeing the ocean battle and inadvertently forcing him to follow. Antony is, in a sense, addicted to Cleopatra and is beginning to make objectively foolish decisions that diminish his honor. 

The phrase "fleshpots" has, over time, developed a sexual connotation, principally suggesting the lower forms of sexual commerce (i.e., prostitution and promiscuity), but the phrase does not appear in William Shakespeare's play Antony and Cleopatra.  The association to sexuality could be a product of the word "flesh," and could be derived from the temptation of the fleshpots discussed in the Bible.  The Biblical meaning, however, has nothing to do with sex or sexuality.  "Fleshpots" in the Bible are just that: pots of flesh or, more precisely, meat.  Moses, having succeeded in bringing the Jews out of Egypt, was consequently and repeatedly confronted by angry hordes of newly-liberated Jews incessantly complaining about the harshness of life in the expansive deserts where food and water were scarce.  Exodus 16:3 refers to one such collective lament directed at Moses by angry throngs:

"Would to God we had died by the hand of the Lord in the land of Egypt, when we sat by the fleshpots, and when we did eat bread to the full; for ye have brought us forth into this wilderness, to kill this whole assembly with hunger."

Shakespeare's play, of course, does not lack for sexual intrigue between Cleopatra and Antony, among others, and the Roman perception of distant Egypt almost certainly allowed for visions of promiscuity that may or may not have conformed to reality.  In that sense, the use of the phrase "fleshpots" would be expected, as the Romans may have viewed ancient Egypt the way much of the world viewed Times Square before its rejuvenation during the 1990s.  In any event, the phrase, as noted, does not appear in the play, but is commonly used in reference to the seedier venues of certain cities and towns.

The "fleshpots of Egypt" were so called because of their loose sexuality and equally loose morals. (In the Bible, the "fleshpots of Egypt" are what the Israelites were trying to flee from due to the overt sexuality, greed, and general sin of the Egyptians against the One True God.) Egypt was all about excess.

Other examples of places being called "fleshpots" are Sodom and Gommorah and Las Vegas, both deemed places were sexuality and morals were/are loose, to say the least! (New slogan: What happens in Egypt stays in Egypt! :)