Why do some instructors care so much about attendance?

Let me start off by saying that it is very unlikely that attendance requirement is purely to protect the professor's ego, as one comment suggested. Any serious educator would understand that the goal of education is not so that the students become increasingly reliant upon the education system. Rather, the goal is to produce students that are increasingly independent, critical, and confident in their own reasoning. If I were a professor, I would be glad that the student can succeed without my help, rather than the other way around. To punish a student for being able to succeed without the help of lectures is simply contradictory to the goal of education.

With that being said, here are some more plausible reasons:

1. The class is discussion based. This is quite straightforward: if you don't attend the class, then you do not learn. The in-class learning experience cannot be compensated by self-study, and exams may not be an ideal measure of such experience.
2. The class meets very infrequently. There are certain classes that meet only once per week. Missing one class means missing a significant amount of work. A related example is science lab requirement. In my undergrad institute missing one lab (without advanced notice) means that you automatically fail the class.
3. The lectures contain information not otherwise (easily) available. This is more relevant for higher-level classes, where there are no standard textbook and the way the professor teaches the material may be unique. The professor may want to make sure that students attend lectures to get the information they need.
4. Culture. In some culture regular attendance is associated with deference to the system and/or the lecturer.

One finally note: contrary to what OP stated in the question, it is my personal experience (in the US) that very few professors would deduct a significant amount of points due to a lack of attendance. Instead the focus, if there is any, is usually on participation of class activity (which, of course, can only be fulfilled if you attend the class). What OP have described seems like rare exceptions rather than the rule.

One theory is that it serves as additional motivation for students to attend class, which in turn helps increase their success in the course. It gives them a short-term incentive to do something which is hopefully also in their long-term best interest.

If attendance is so crucial to doing well in the class, wouldn't the students who don't attend do poorly in the exams anyway?

In many cases the instructor has found from experience that this is true. But the student (who has less experience) may not be as convinced.

Consider a student who wakes up in the morning and doesn't feel like going to class. In a class with no explicit attendance requirement, the student may rationalize: "I will just study harder tomorrow to learn the material that I missed, and I'll still be able to do well on the exam, so skipping class will have no consequences." But they overestimate their ability to do that, and end up not learning it as well. Or tomorrow they put off the studying until the next day, and so on, and fall behind. As an eventual result, they do not do well on the exam.

In a class with an attendance requirement, the student knows for sure that not attending class will have negative consequences. The biggest consequence (failing the exam) is very likely but not guaranteed, and the student may not be able to impartially evaluate just how likely it is. But loss of attendance points is guaranteed. So the student cannot pretend that skipping class is harmless. Thus the student is more likely to actually attend, which is in their long-term best interest anyway.

Hopefully, the ultimate result is that a higher percentage of students are able to meet the standards of the class. The flip side is that some students who have good attendance but poor performance otherwise may get better grades than they "deserve", but the instructor may feel that this tradeoff is justified.

So here are some observations from my perspective: I teach at a large, urban, community college. We are open admissions (no starting prerequisites) and the students have many challenges (graduation rate 15-20% in the university at large; ~25% for our own college). I'm very much an outlier in that I'm one of the few faculty who don't want to be tracking attendance closely. I'm constantly trying to understand why other faculty are so adamant about this; and frankly I have yet to receive a super-coherent account of it. But some bits and pieces that I get at times:

• Students may be so weak that they are subject to the Dunning-Kruger effect; they have no idea how in trouble they are, or what it takes to remediate their weaknesses. Perhaps they are not in a position to make a rational choice about their academics, and at this point need some enforced guidance in that regard, esp. in a linked-knowledge STEM discipline. (To me, this is the strongest argument, the one that allows me to at least entertain the thought once in a while.)
• There may be a legacy/cultural aspect; for example, at our school we are given paper rosters with calendars marked out on them for each class, with the direction to mark it a certain way for attendance every day. I've never seen a contractual/handbook requirement that we do this, but the paperwork says so, and they are required documents to be filed at the end of the semester.
• There may be institutional reporting metrics at stake. For example, if a student misses 4 classes (course meets twice a week), then the college lets us drop them from the course, and my department quasi-mandates that we do so. I think part of the reason is that the student then counts as an "unannounced withdrawal", which makes our "failure" statistics look better (to very high-pressure stakeholders higher up in the university administration).
• Some instructors may do this to make the course easier. I've heard at least once that an instructor in another department had, say, a 70% grade component based on attendance. That is: a student is not required to perform any work whatsoever; as long as they are physically present, they can pass the class (and thus relieve some amount of pressure on the instructor, I presume).
• "Remember, the attendance rosters are legal documents. Years ago there was a student accused of a crime. They were proven innocent because of their being marked in class that day, which counted as an alibi." (I've heard this lore multiple times.)
• "Attendance is important to reporting for financial aid; we must confirm that students are attending for certain financial aid requirements." (About 75% of our students get federal/state grants?)
• "Don't you think attendance is important? Don't you want students to succeed?"