1) @alarson was also talking about college. That chart is showing data for primary and secondary schools.
2) "Non-teaching staff" does include a lot more than the dreaded administration. I bet a lot of that increase is from schools hiring ELL people and teachers associates. The change in strategy of keeping kids with special needs in the classroom with their peers instead of in their own room necessitates more assistance for teachers
I was not trying to refute his point entirely -- some of that growth, even in college (e.g., adding administrative personnel to professionalize some things that faculty used to do themselves to "pitch in" for the department or college, such as admissions or finance, or adding IT personnel with the rise of personal computers) makes terrific sense.
I do not think there is one answer here. I would imagine reduced state support is probably the largest reason just thinking through the numbers. Increasing "school services" (i.e., for programs and services that did not exist in the university of the 1950s but do now, maybe call it "administrative bloat" depending on who you ask) is probably more of a contributing factor, but one that I imagine has to have some influence.
Then again, you know me, I think we more-or-less over-invest in education as a society anyways, mostly because education is a signaling game.