Here's a piece that is touched on in a few other answers but which seems to be unstated: the United States' university system has many of the worlds top schools because it is massively unequal.

The US has more than 4,000 colleges and universities — the "best" universities may be in the US but so are some of the worst. These two facts are not unrelated. The US higher education system is more effective at concentrating resources (top students, money, grants, and the best faculty) in a small number of universities at the top than any other national system.

In much of the developing world and in many small countries, there are simply few resources. In many developed countries, universities are funded by governments that attempt to distribute resources widely and fairly (if not equally) across universities in their systems. If, for example, most students go to universities near their hometown, it seems unfair to send the best faculty and most of the grant funding to the university that trains the folks who happen to live in the capital. In the US, there are few pretensions. Most of the top US schools are private schools that compete with funds and that have a massively disproportionate share of the money, grants, top students, and top faculty. If you happen to come from a state with a poor university system, it's just bad luck.

Because there are few checks on inequality in the US system, this leads to a feedback loop where universities at the top use their resources to capture more — aggravating inequality over time.

Here's a piece that is touched on in a few other answers but which seems to be unstated: the United States' university system has many of the worlds top schools because it is massively unequal.

The US has more than 4,000 colleges and universities — the "best" universities may be in the US but so are some of the worst. These two facts are not unrelated. The US higher education system is more effective at concentrating resources (top students, money, grants, and the best faculty) in a small number of universities at the top than any other national system.

In much of the developing world and in many small countries, there are simply few resources. In many developed countries, universities are funded by governments that attempt to distribute resources widely and fairly (if not equally) across universities in their systems. If, for example, most students go to universities near their hometown, it seems unfair to send the best faculty and most of the grant funding to the university that trains the folks who happen to live in the capital. 

In the US, there are few pretensions. Most of the top US schools are private schools that compete with funds and that have a massively disproportionate share of the money, grants, top students, and top faculty. Because there are few checks on inequality in the US system, this leads to a feedback loop where universities at the top use their resources to capture more — aggravating inequality over time.