Humans (and presumably human rights, if assuming they're fundamental) existed before countries. So how does a concept that came later (a country) suddenly trump something that's fundamental? What is the moral mechanism by which a group of people declare a certain area a country and suddenly gain property rights over everything within that area?

Human rights, just as countries, are a human invention. These inventions are extremely useful, because they allow some rules for conflict resolution — without them, it's just rule of power and violence.