The real explanation for rising prices amid greater homes per capita is actually quite simple, fewer kids. Kevin Erdmann has an excellent post on this going through the numbers in detail. I will illustrate with a stylized example.
Suppose we have 100 homes and 100 families, each with 2 parents and 2 kids. Thus, there are 100 homes, 400 people and 0.25 homes per capita. Now the kids grow up, get married, and want homes of their own but they have fewer kids of their own, none for simplicity. Imagine that supply increases substantially, say to 150 homes. The number of homes per capita goes up to 150/400 (.375), an all time high! Supply-side skeptics are right about the numbers, wrong about the meaning. The reality is that the demand for homes has increased to 200 but supply has increased to just 150 leading to soaring prices.