How It All Goes Down
We have to admit; we start out feeling pretty bad for Strepsiades. The poor guy is apparently in mountains of debt because his son, Pheidippides, has a fondness for horse racing. So, it's not even his debt, but his son's. Cue the violins, because we're so ready to sympathize with his plight.
However, that feeling doesn't last super long. It seems that Strepsiades has no intention of paying his debts; instead, he wants to find a way to argue his way out of them. Hey, we get that it wasn't exactly his behavior that got him into that mess, but that's not cool—if you borrow money, you should pay it back; that's just Good Citizenship 101 (unless horse racing offers debt forgiveness programs, of course).
So, he decides to get his son enrolled at the local "Thinkery," a fictional school run by the (non-fictional) philosopher Socrates, so that Pheidippides can learn how to argue their way out of the debt. (Yup, Aristophanes inserts a real person as a character in his play, and let's just say that the portrayal is not terribly flattering… but we'll get to that in the "Characters" section.) Unfortunately, his son is not really that interested in helping, so Strepsiades has to go enroll himself.
Even more unfortunately, Strepsiades is a pretty subpar student, and Socrates finally throws up his hands and drops him. Not to worry, though—Strepsiades finally convinces his son to enroll and give it a try. And it works; Pheidippides ends up learning everything he needs to know about the "Worse Argument" to argue effectively against his father's debtors.
However, he uses his newfound knowledge to do some other stuff—like justify beating his father. Yup, when Strepsiades gets mad at Pheidippides at a feast celebrating his accomplishments, Pheidippides starts abusing him physically and verbally, and then adds insult to injury by using his "achievements" to justify it all.
With that event, Strepsiades finally realizes that perhaps he made a mistake in thinking that it was okay to use logic and knowledge to achieve immoral ends. So, he formally rejects the "Worse Argument" and sets the Thinkery on fire to punish Socrates and presumably put a monkey wrench in their ability to teach that kind of dangerous stuff to others.