To speak nothing of the professors...
Higher ranked schools are often better at research as well, you can get more bang for your buck than lesser ranked schools.
Note that this depends on what you want for "bang". At less well-heeled schools, a far more modest donation can steer departments etc.
Just to be clear, Princeton ranks close to 15-20th in CS. Many state schools do out rank Princeton in CS and they do not have such labs yet.
I can understand if IAS or the Math department was the main reason to open a lab there. But, if it was quality of CS faculty that exclusively factored into the decision, then there would be other state schools in the line ahead of Princeton.
This probably had a lot to do with it. Schools like that have huge endowments (Princeton's is north of 20 billion dollars), and as such can afford to attract very good faculty. Very good faculty can lead to very bright students.
Princeton is also going to have a lot more defense dealings than something like a state school and where cutting-edge tech is being researched, OGAs are guaranteed to be sniffing around and weaseling their way into the research with funding.