Well, an important consideration with Lysistrata is the nature of the comedy. A modern reader's first impression normally centres around how strange a tactic the withholding of sex is for political reform. I'm sure the ancients though that this was funny as well, but remember that this play was written as a critique of the Pelopponesian war. The punchline, in the end, was not the humorous way in which the women enacted change, but that it was the women themselves who had to act for this change.

Greek views of women's intelligence were not flattering, and for even the WOMEN of Greece to see that the men were being fools would have been a very serious critique of the leaders of both cities. If we look at ancient greek drama, even slaves are sometimes allowed insight, but serious intellectual clout is almost never attributed to women.

If I was home, I'd have a source, I'll see if I can't find something in the next few days.

For us its a funny story of female empowerment. But I think the Athenians would have seen this as a very black comedy. Athenian (and by large all Greek) opinion on women was fairly misogynist. One gets the impression from the literature that if men could have babies, they'd ditch the whole women thing altogether.

Now aside from the sex strike, the older women (and in ancient comedy, older women were always undesirable for sex) were sent to seize the acropolis to prevent funds being taken out to pay for the war. When a magistrate comes to take money to pay for oars for the navy and is refused - the women are going against the democratic will of the citizens who voted for that money to be spent. The women are defying the will of the people.

Furthermore the seizure of the acropolis would bring back memories of the Spartans who seized it and were trapped there after trying to install an oligarchy.

And the women aren't trying to change society, to make things better for women. They're trying to get the men home and a return to the status quo before the war. So it's hardly a feminist flagbearer. Worse, near the end Lysistrata tries to show how the Athenians and Spartans helped each other before but listed two of the worst examples for justification:

• That the Athenians sent an army to help the Spartans during a helot rebellion - forgetting that the Spartans were suspicious of the Athenians and sent the army back. This was a huge insult to Athens and ensured the pro-Spartan Cimon was ostracized. This left the anti-Spartan politicians (Themistocles, Pericles, Ephialtes) to shape Athenian policies which would lead to the war.
• That the Spartans sent an army to help expel the tyrant Hippias - forgetting that immediately afterwards the Spartans tried to setup an oligarchy and were blockaded on the acropolis until they agreed to withdraw from Athens.

So this demonstrates that Lysistrata is a political idiot. But the men aren't paying any attention to her anyway since they're eagerly oogling the personified Peace, she's just flapping her gums talking nonsense.

To make a crude analogy, imagine if in the early 2000's when both the Iraq and Afghan war were ongoing and Iraq seemed to be falling apart, someone put out a movie where illegal immigrants refused to work the low-paying jobs so urban life would grind to a halt, but also seized the federal treasury to prevent further funding of the war - all to force the government to bring the soldiers home from the war.

Aside from the matters mentioned in the answers that have already been given, I think it's wise to stress the topical nature of the play -- that is to say, how it was designed around current events. At the time, Athenian comedies were written to be performed on a specific occasion, as a one-off component of a specific festival. We don't know the details of the festival at which Lysistrata was performed, unfortunately: it was in 411, the same year as Aristophanes' Thesmophoriasdousai, so presumably one was performed at the Lenaia festival and the other at the Great Dionysia, but we can't be certain which was which; both festivals were in winter/early spring (January to March), at least, so we can be sure it was before the oligarchic coup d'état later in the year, and the mutiny of the fleet at Samos. We also don't know whether Lysistrata came first, second, or third in its year. For all we know it may have been a flop at the time, even if it became a classic in later centuries.

As an Athenian, you would be keenly aware of just how big a disaster the Peloponnesian War was turning out to be for the Athenian state. Twenty years earlier Athens had seen itself, with some pride, as the leader of a commonwealth (or empire, depending on your point of view). About 18 months before the festival where the play was performed, Athens had suffered one of its worst-ever military disasters -- on a par with the sack of Athens in the Persian Wars -- when it sent its entire fleet to Sicily to plunder the island for funds to support its campaign against Sparta. The entire fleet was lost. This was absolutely catastrophic, and would cost Athens the war, though they managed to hang on for another seven or eight years. The hawks in Athens' political sphere managed to throw together another fleet to protect itself in the meantime, but it was at great cost to the city in both economic and personal terms.

Aristophanes hated this. Everyone hated this. Lysistrata was, in effect, his answer to the hawks, and in the audience, you would realise this at once. This play was the kind of thing that could start riots: Aristophanes' casting of women as the heroes of the day was his way of softening the play. After all, no one would be able to take seriously the idea of women starting a sex strike and barricading themselves inside the city's treasury... This choice of plot ensured that no real, specific individual was being targetted for criticism.

You would also be aware of many other topical references, of which the most important is probably Lysistrata herself. She is a parody of a real person in Athenian public life: the priestess of Athena Polias ("Athena of the City"), Lysimache. The names even mean very similar things: lysi-machē "she who breaks up fights", lysi-stratē "she who breaks up the army". Lysimache was well known, had been in office for several decades (she was an elderly woman; this may help explain some aspects of Lysistrata's characterisation in the play) and, like Lysistrata, was a formidable opponent of the war. (That being the case, there was still no criticism involved in Aristophanes' parody of her: Lysistrata is more a homage to Lysimache, not a satire of her.)

There are many other topical references too, though not as many as in some of Aristophanes' play (the Lysistrata has no parabasis, which is a choral ode where the Chorus steps out of character and starts giving explicit political advice to the citizen body). Naturally the culture of humour is different in some ways too, but the topical nature of the play is an absolutely key element to your question.