You are reading a play by one of the comedy masters. Molière is one of the most well known and most praised comedy playwrights of all time.
Although Molière was great at all kind of ways of making you laugh, his specialty was something called the comedy of manners. What's that, you say? It's a comedy that focuses on making fun of the manners of certain (elite) social classes.
Back in Molière's day, that could mean making fun of the nobility and the rich for their hypocrisy and lies. In our times it might look something like Seinfeld or Woody Allen's movies—or an elite "expose" like Gossip Girl.
It's pretty clear that Molière is having a great time making fun of the social rules surrounding upper-class Parisians, like when Clitandre modestly says, "I'm clever, handsome, gracefully polite;/ My waist is small, my teeth are strong and white;/ As for my dress, the world's astonished eyes/ Assure me that I bear away the prize" (3.1.7). Or when Oronte stutteringly reads his poem and then gets offended when Alceste doesn't like it.
In the end, Molière lets us know how silly it all is because everything come back to bite Célimène in her ruffle-covered butt. Poor little rich girl.
The same things that make The Misanthrope a comedy of manners are the things that make it a drama and a satire. Because comedies of manners are more concerned with making fun of some social class than they are with, say, action sequences, they often focus on some conflict that riles up all of the characters' emotions. That's the dramatic part.
Comedies of manners also make the social classes they are making fun of seem way worse than they are. There might be some truth at the bottom, but you will never actually meet these people because they have had all their bad parts amplified times ten. And the point of all this is to convince the people watching—probably aristocrats—that they should, you know, get their acts together. That's the satire part.
And one more thing. Normally comedies have happy endings, and, if we're talking classic comedy, they're even supposed to end in marriage. The ending of this play is all kinds of weird and just ends all of a sudden. Sure, Philinte and Éliante get together, but Alceste is on the lam, headed for the woods, and suddenly has no girlfriend. That's actually kind of a bummer.