Demand does go up, because people who earlier couldn't compete for the good we're talking about now can. There are x locations each night to sleep that are zone

Demand does go up, because people who earlier couldn't compete for the good we're talking about now can. There are x locations each night to sleep that are zoned for residential purposes, some more attractive than others. This is the supply. There are y people competing for those; y is roughly equal to the amount of people who have their primary residence there. Now y' people who are not primary residents start to compete for those places. Result: demand is y + y'. X remains constant (in the short term). Result: P goes up. The 'cheaper' meant compared to the prices of another market, i.e. the hotel market. This is a cumulative effect, on top of the first-order effect of those consumers being let into the market in the first place.