How does this happen? Is there a law which just gives you a citizenship if you stayed for N years?

The exact country isn't clear, it depends from country to country. Spain for example have "arraigo social", where I think if you've stayed for 3 years (illegally/legally) and can demonstrate you've ended up in some sort of "link" with Spanish society (like having a permanent job) you can apply for a "temporary residence and work permit". Once you have that, you're legal and you could apply for permanent residence and eventually citizenship, granted you fulfill those requirements.

I have a bunch of friends, with jobs ranging from bartenders to software developers, who've successfully were allowed to stay in the country after doing things that way, initially staying illegally and later regularized their situation.