Here's how it went down. First, Germany slaughtered and exterminated millions of people, invaded tons of countries, destroyed their infrastructure, commited heinous crimes, etc. What we know as World War II.
After the end of the war, West Germany, as a valueable ally of the West against USSR, got all kinds of help and special treatment to rebuilt its economy. And they started a program for bringing in workers from places they invaded and decimated, like Greece. Those workers, like the involuntary "arbeit mach frei" workers before, were brought to serve as cheap immigrant labor, under exploitative conditions, to assist the very people that had unprovoked invaded their countries and killed their relatives and fellow citizens a decade or so ago. The guest-workers, besides being exploited with lesser wages than Germans and fewer opportunities for advancment (like b-rate citizens) they were also treated with extreme racism and prejudice.
Now, the problem of the Germans (the "they overstayed their welcome") was that these people didn't just disperse when they were done with them, but instead felt that those hard working years in Germany gave them a right to stay and be integrated as full citizens.
Such nerve, huh?