First of all, Aristophanes' Lysistrata is a drama. We're not talking about drama in the sense that made-for-TV movies on serious topics like WWII or forensic science are dramas—Lysistrata is definitely not going to bum you out. We're talking about drama in the sense of drama class in high school.
That's because Lysistrata is a play. It's meant to be performed—although, given the kind of stuff that happens in Lysistrata, it's unlikely that any principal would allow Lysistrata to be performed in drama class. It's too hilariously dirty for that.
The keyword is hilarious—from the get-go, Lysistrata is a comedy. It features a loopy plot with unlikely events (women going on a sex strike to bring an end to war), lots of humor along the way (in the form of sex-puns, smutty jokes, and very awkward situations), and it all finishes with a happy ending, er, we mean "pleasant conclusion."