He says it will only take you so far in explaining it. I've had this conversation on a few separate occasions on facebook recently with people (I assume Brad is not of this bunch) that thinks federal subsidies are driving education prices rather than vice-versa.
Demand for education is largely driven by wage premiums between different educational attainments. It is a consumer good too, and if that were the whole story then we'd expect the difference perhaps to be dictated solely by things like Baumol's disease. But I think with secular technological development you have a continuously increasing demand for this service because you have continuous upward pressure on wage premiums, and on top of that (as Brad points out) we probably haven't had any tremendous productivity gains in delivering that service. That is a recipe for rapidly rising prices.
Finally, in the last decade or two you have to take account of the fact that we are educating a sizeable chunk of the foreign labor force.
And as Archibald and Feldman point out, increasing income plus reduced costs for other goods and services over time leaves us with nothing to spend our money on except for eds and meds.