In the beginning of "University Days," Thurber discusses his problems with botany class at Ohio State University. Thurber had trouble seeing through the microscope. (In real life, Thurber had poor eyesight). The professor is not helpful, choosing to berate the narrator (Thurber) rather than try to understand what the problem is. The narrator takes a deferral and retakes the class only to shift the lens so that it reflects his eye. The story satirizes the narrator's own confusion while mocking (and maybe even sympathizing with) the professor's frustration.
The narrator then describes his problems in economics. Having just come from botany, he'd get the two mixed up. The fact that he would get botany and economics mixed up is laughable. It is probably a combination of his own frustration and (perhaps more so) bad teachers.
In economics, the narrator presents an even more confused student, Bolenciecwcz, a tackle on the Ohio State Football Team. The narrator describes Bolenciecwcz's situation:
In order to be eligible to play it was necessary for him to keep up in his studies, a very difficult matter, for while he was not dumber than an ox he was not any smarter. Most of his professors were lenient and helped him along.
Subsequently, the professor asks Bolenciecwcz to name a mode of transportation. He can't. Therefore, the professor and some of the students make train noises; the kind you would make to get a toddler to say "train." This is comical and satirical so it's light-hearted, but it's also an overt statement about the lack of academic integrity at the university. If players this unintelligent are admitted to a university, there is clearly a relaxation of academic requirements for football players.
The narrator then describes his problems in gym class. He is not allowed to wear his glasses in gym, but can't see without them. Caught in this catch-22, he gets a friend to pass the swimming requirement for him. So, the university makes it easy for the football player to get by academically but they won't allow the typical student to wear his glasses. Ironically, using his friend to swim for him is a "visual" deception.
The next example is when one of his fellow students doesn't know what college he was in, such as Arts or Engineering.
"What college are you in?" the instructor snapped at the youth in front of me." Ohio State University," he said promptly.
The narrator makes another appropriately mocking statement about the military requirement. "We drilled with old Springfield rifles and studied the tactics of the Civil War even though the World War was going on at the time." The university has them studying war tactics nearly 60 years old at that time. The General was so incompetent that he was more interested in killing a fly than he was in his cadets.
This story is a satire so it uses humor to mock and criticize the university's flaws. Satire is often done humorously but tends to have a critical agenda. In other words, sometimes the goal is to call attention to flaws in order to inspire or suggest improvements.