The "free" Bachelors is still mostly the privilege of American students (with a few truly exception international students sprinkled in).

For PhD, As sidww2 pointed out, and what I have experienced as well, the admission pool becomes much more international; and now you are competing with the world. There is a phrase about China saying that underlines the scale well: "in China when you are one in a million -- there are 1,300 other people just like you".

Moreover, overall the number of spots in graduate schools is by far smaller than for undergrad; making competition even tougher.