To try to get back to the OP's premise, though exaggerated as his usual posts with AK-47s and Pelican Bay Prison, etc, I think as a generality in USA this excuse is used to not make relationships reciprocal in nature. As in, because nobody "owes" anyone anything, someone can do something nice for you, and you're not obligated to do anything back in return. So friends in USA I find quite often do not return favors, and I think I myself have been guilty of it, too. Obviously not every action you do or others do has a reciprocal indebted forever kind of thing, but even simple things like, if my neighbor mows my lawn for me, I try to say, get him a case of beer in return, etc. In Japan for example, it was custom (can Yohan chime in to see if this is still a thing?) to try to get your neighbors small gifts, and it ends up often being a sort of weird loop of neighbors buying gifts for each other forever each side feeling they "owe" the neighbor back. Without this reciprocal kind of relationship in relationships, then ultimately relationships become sort of meaningless because anyone can flake at any time for any reason, and in general relationships are hard to form because you cannot make the first move of the "give" in the reciprocal loop and expect anyone to care to reciprocate back, because of the "nobody owes you anything" mentality.
As far as in general with US relationships, as in, sexual or romantic, but I guess even broadly politically speaking even in terms of welfare or really anything of any public good at all, the term gets thrown around again. "I'm too poor to afford drugs I need to live." Well, hey, nobody owes you anything! I remember prominently a US politician writing a letter back to a constituent who asked what he could do to help her afford her Epipens and insulin pumps she needed, and the politician replied back with basically "nobody owes you anything, free market, haha!"
http://www.foxnews.com/health/2016/06/3 ... ghter.html Though unrelated, when someone asks on a normie forum about their lack of a girlfriend/etc, the canned reply generally someone throws in is this "nobody owes you anything!" thing. Instead of at least remarking with some sort of sympathy for a bad situation to them, like the insulin letter above, there's just that, which is totally unhelpful and meaningless to try to justify a situation that is completely wrong, like modern dating often can be.
This sort of basic idea of "nobody owes you anything" is I think one of the things most wrong with USA, it's a saying coming from a very hyper individualist and hyper competitive angle.