From the various EU countries I lived in, I would say it’s a mix of:

• Residential housing being transformed into tourist accommodation, i.e. AirBnB. This has both cut down the Offer side in the residential market and increased Demand, pushing prices up.
• The ultra-low interest rates that were the supposedly temporary fix for the 2008 crash led prices to go up because lower rates means the same monthly mortgage payment can cover a larger mortgage so ultimately those lower rates just made mortgages go up until people were paying the same each month as before with higher rates.
• Residential units became investments much more than before, especially when the reactions of Governments and Central Banks to the 2008 showed that they would support the Real estate Market no matter what, flooding a market which before was mostly owner occupiers with buyers with much stronger financial capacity (all the excess money the wealthiest accumulated in the last couple of decades has to go somewhere). This had increased Demand plus put far poorer common locals in competition with International wealth.
• Building has fallen, especially in places were before there was a lot of social housing being build by the State. In the neoliberal age the State refrains from “intervening in the Market” which in this case means from competing with builders by building residential units. The result is a Market were the builders themselves build less in order to make more because a shortage of housing pushes prices up. This too cuts the Offer side.
• A change in the typical size of households, with much more people living on their own in singled person households than before. This increased Demand, even in countries with no population growth.

This has affected poorer EU nations as much as richer ones, so somebody’s suggestion that it’s to do with freedom of movement is the same kind of nonsense I heard from Brexiters in the UK: my native Portugal has a massive housing bubble even though things are so bad that the country loses half of its university graduates every year to emigration, which together with all the rest means almost 200k people leaving every year (edit: out of a population of around 10 million, so 2%), and yet the average age at which a young adult in Portugal leaves their parent’s home is now 34 years old. Literally decades of Government measures incentivising house price increases (from a refusal to properly regulate conversions of residential units into AirBnBs and actual incentives for rich foreigners to invest in PT property by giving out Golden Visas in exchange for such investments, to a complete collapse in the building of social housing by the State) are exacerbating a brain drain as well as making the population aging problem worse since even those who stay have children later and have fewer of them because the insane costs of housing mean they can’t afford to have children earlier or more than 1.

Politicians in the current political system love rising house prices because, due to the way GDP is calculated using an Inflation metric that understates the effects of house price inflation, house prices going up translates to a higher Official GDP figure - I.e. GDP “Growth” - which is politically treated as representing an improvement for the country and the people in it.

I would say that the root cause of the problem is Neoliberalism as well as crooked politicians who themselves are high middle-class or wealthier, hence housing investors, hence gain directly from doing all they can to prop-up housing prices.