The share of young(ish) people living with their parents varies considerably across Europe:
In many countries the trend is driven by culture.
The above chart is from this piece over at Quartz. In describing how the numbers have changed over the past five years, the article notes that for example the number of 25- to 34-year-olds in Spain has plunged by 16% over the past five years - many have left the country.
In addition to culture and current economic context, very different welfare systems (mainly in terms of unemployment benefits) are also a key factor here. Welfare systems vary significantly between countries, especially in terms of eligibility and qualifying period, coverage rates and duration.
For example, Denmark is one of only five countries in Europe where the self-employed have the same entitlement to unemployment benefits as employees. You can find comparative details of different systems in Europe at the end of this report (PDF) and using the MISSOC database. Furthermore, even though tuition is free, students in Denmark have access to an extremely generous (and universal) grants system, which provides about €750/month for those not living with parents.
A more homogenous variable is the link between employment and those staying at home. The following maps using Eurostat data show recent unemployment rates among 25-29 and 25-49 year olds (I was unable to easily find data for 25-34 year olds only).
While not the only driver, there is a clear link between the unemployment data and the numbers still living at home with their parents.
There has also been a shift in terms of type of employment. In average, between 2007-2012, there was a >2% increase in the proportion of employees in temporary jobs.
More than one in three of those living with their parents is in temporary employment.
For most people this isn’t out of choice.
All this has played (or is likely to over time) a role in stagnant or falling disposable income in several countries. While in other countries, where median income has increased, such as Estonia, Latvia and Slovenia, the numbers living with their parents have fallen.
Young adults aged 20-34 living with their parents in the UK, 1996-2013
Charts via eurostat and this UN report.
In conclusion, to answer the question - it’s a combination of culture, multiple economic and labour factors, and welfare systems - but not necessarily, nor always, in that order: it’s extremely complicated. While every country has specific and diverse characteristics, and culture and welfare systems in many ways act as a "baseline", more recent labour market trends are as significant as mamma’s cooking.