The feeling of inferiority entering a PhD program due to one's undergraduate school rank

(Source: I am a math professor at Stanford, with connections to the CS department and many former students in the CS Master's program, and experience with graduate admissions.)

Nothing about this situation suggests that you are any less intelligent than your classmates. The admissions committee isn't running a charity: if they thought you weren't prepared to excel, they would have admitted another student from MIT instead of you. They invest a lot of effort into determining which students are positioned to be successful; it's not a perfect system by any means, but I think they know better than you do whether you're prepared.

I have never considered undergraduate institution or background when considering who to work with, and I haven't heard of any colleague doing so either. As a grad student you get better and better at learning, so it's easy to patch any holes in your background.

If you are concerned that you haven't had the opportunity to take all the classes that the CS undergrads at Stanford have: well good news, the majority of CS courses are online, including all the lectures, slides, and homeworks. And you have 7 months before you'll start grad school this fall -- that is plenty of time to complete all the courses you're worried you've missed, if you really feel that's the best use of your time.

This was me years ago! Tom Church's answer is a good one, but here's my idiot opinion: yes it's super intimidating and my first year I was sure I would get kicked out. Part of it was undeniably the fact that my lower ranked institution really didn't offer the same/as many classes as what my colleagues from MIT and Caltech got, and I really was less prepared. But part of it was also that those colleagues were extremely confident, having come from those top ranked institutions (and also had some of the garden variety arrogance that seems to come standard with STEM majors, which amplifies that confidence). What's more, the grad classes were extremely difficult and some of my colleagues weren't as used to having to struggle to understand things. But annnnnyway- most of us did fine and graduated (including me yay). Some- including some ivy leaguers- dropped/failed out. The main difference was in work ethic, determination, and being a personable non-asshole (you'd be surprised how far that last one will get you).

I did undergrad at a non-R1 private university that does well but not excellent in national rankings. I did my Ph.D. at the top school in my discipline, and almost all of my classmates had undergrads from R1 public schools or Ivy-caliber private ones.

If anything, my experience showed me that the fever surrounding admissions to undergraduate institutions is way overblown. I was well-equipped and never felt out of my league. The difference between a Harvard or decent public school undergraduate degree is less than you think.