There are two things going on that would cause a housing shortage in the immediate postwar. One is lack of housing starts during the war in places people would work after the war. The second is disruptions in living situations that people wanted after the war.

Converting to a war economy involves a lot of people moving around. The huge factory at Willow Run for producing B-24s required construction of 1,900 dormitory housing units, and it was by all accounts immediately full as thousands of workers poured into the Michigan town from all around the country. After the war, when Willow Run's demand for workers fell dramatically, the people living in those dormitories had to move somewhere else to find peacetime work.

Additionally, during the war, housing construction not related to war work was effectively forbidden by the rationing system and draft. So the housing starts that would have normally happened in peace time were not built because there were no workers available, no lumber available, perhaps no nails or screws available without authorization from various goverment control boards. Basically the civilian economy got along with what it had in 1939-40, or what absolutely had to be built for the war industry. So basically there is 4+ years of pent up demand and no additional housing supply.

When people want to move back to a civilian job, or want to move into a home big enough to fit a couple starting a family, it's very hard because there have been four years of building dormitories near war factories, not family homes near peacetime work.

Source: Maury Klein, "A Call to Arms: Mobilizing America for WWII"