Why do universities require international students to show language proficiency?

The short answer is that when people want something, they often experience cognitive distortions that prevents entirely accurate assessments. Dunning-Kruger is a good example of one of these effects, where a student thinks, "Oh, I'd love to study in Paris - and lots of people speak French, so I'm sure I could pick it up!" Learning a new language to a college-level of proficiency in reading, writing, and speaking is incredibly challenging, yet some people inevitably misjudge this.

Sure, the student would eventually figure this out and face the consequences, but in the mean time there are lots of reasons the institution wants no part in such a situation. For one, it's sad - a student will struggle, feel isolated, and most people end up blaming others for their situation and don't take complete responsibility ("why did they let me in if they knew it wouldn't work?!"), so it can be really unpleasant. I saw a student from China try to make it in English-only classes, and I couldn't make out if he even understood more than a few words in English - he only ever nodded, looked confused, and looked back down at his paper. It was just sad to see, and I can't imagine how this was helpful to him - the only way he could succeed was to have an interpreter (he didn't have one), or cheat, and that's just a terrible situation to be in.

Most institutions also have various statistics like drop-out and graduation rates, as compiled by an office like Institutional Research, often reported to the government (for "public" institutions, and others who take government money). Institutions have a lot to lose for regularly taking on students they have a very good reason to think will fail in their program, or will require disproportionately high resources to support. So they want to try to avoid such situations, when possible. Finally, positions in a class/program/institution are often limited, and accepting one student can mean necessarily rejecting some other student, which makes administrators even more keen to be careful of offering a spot to a student unlikely to succeed.

A language proficiency requirement is no different from any other proficiency requirement. Universities will typically ask you to prove your subject proficiency through grade boundaries in lower qualifications and require you to present proof of this rather than simply accepting any student who thinks they are good enough.

The reason for this is simple: it is difficult to effectively teach a class to students who do not meet consistent expected levels of competency. The absence of enforced standards means that there will be am increased number of people in the class who do not meet the required level of competency and thus effective teaching will be limited.

The accepted answer is good but from an analytical point of view a large language barrier is very bad for any university. You want a student's grades to reflect the area that they are studying, not their basic language ability.

I went to a US school that had an open program with Spain. Some of the students that came over had great English skills. But the ones that didn't either did really poor or were given (way too much) leeway in courses. I remember reading some of my roommates papers that he would get a low B or a C and I was like "what the hell!" I mean just jumbled English to the point where you weren't even sure what point he was making.

Now he was a smart guy. But in these same classes I would get a paper back with markings for super minor grammatical edits, to the point where the teacher is asking for a different like word. So the language barrier has a huge impact on any class unless it is pure mathemtatics/programming and even in those fields you could mistake a questions point.

So simply a university wants to measure how a student did going through their courses. They cannot accurately do this if that person can't convey the language correctly. And on the opposite side of this the university doesn't want to flunk out smart students that actually know the subject matter because they can't express it right.