Most PhD programs at American universities cost nothing for most students, i.e., if the university accepts you into a doctoral program it almost always comes with full tuition remission (i.e., you pay nothing) and a modest stipend (i.e., they pay you, somewhere between $10,000 and $25,000 per year). The stipend can be in the form of a teaching or research assistantship, in which case you will work 10-20 hours per week helping a professor, or in the form of a fellowship which requires no work (but fellowships are highly competitive and you likely would not receive one).

Does that change if you're an international student? I honestly don't know, but it would surprise me.