There are a lot of factors and I didn't meant to suggest the current state of affairs is entirely a result of decreased state funding. But no doubt it plays a huge role and just as importantly has driven a lot of behaviors and decisions on the part of universities in that time.
As for private schools, yes their tuition has increased as well. But studying private school tuition is meaningless without paying attention to the discount rate, and for most schools that is way up as well. So it's not quite what it seems for private schools.
I think there are probably six factors, thinking through it.
(no particular order here)
1.
Baumol's cost disease -- This is an old problem in economics. Service industries are not becoming more productive very quickly. It takes as long to grade an essay now as it did in 1950. At the same time, other sectors are seeing massive productivity increases and able to pay higher and higher wages. In order to attract workers, service industries have to do the same thing, which inevitably leads to higher prices because they do not have the productivity increases to hold them down. The same issue happens with healthcare with its costs over time.
2. Universities used to be faculty and... that was it. The mass professionalization and bureaucratization of educational institutions comes with its conveniences but costs, too, as well as state- and self-imposed mandates for certain service programs.
3. Students expect more -- the monastic experience used to be part of the deal, especially when parents were mostly paying for school. As students have started to self-finance, they have started asking for more, and hence dorms have become nicer.
4. IT costs -- computers and electronics for students and, perhaps underrated, the costs of equipment for faculty have gone up a lot. That has to be paid for. I would imagine research professors in engineering, sciences, and agriculture need a lot of $$$.
5. As the returns of a degree for individuals have gone up, so has willingness to pay. Their ability to finance more and more debt allows more to be charged. Colleges are simply collecting some of the windfall of the students' future wages and ability to finance for themselves.
6. State contributions to public schools have declined.
I think they all probably play a part.