A big part of the social contract behind society is something like "the working class agrees to respect the private property rights of the capital class, in exchange society ensures that they are able to maintain a decent standard of living". Many people consider owning a home part of "a decent standard of living" both due to the economic stability it provides and due to the autonomy it provides.

Another way to look at it is that a home is the primary large piece of property that most people own. Therefore, owning a home gives people a stake in a capitalist society.

What both of these are getting at is that if common people are unable to buy a house then they are less likely to feel obligated to support the rules that society is built upon.