From what I have seen, international students are a big ticket item for the colleges that have the gears in motion to keep them and attract more. My Undergrad for example had about a 10% population and a large proportion of them were either from Asia, Europe or Middle East. Some colleges also have what colleges call an American English Institute, which in some ways creates partnerships with other colleges overseas and helps those colleges to consider sending summer groups to experience college life first hand. Some of those students come back and tell friends. When you have a group of visiting Phd candidates and faculty, the word that spreads has more weight.
Regional considerations fall behind that of money. It's likely that an international student won't transfer (i have no idea what that is like now, but one feeling amongst undergrads was the whole reporting your status...)So once you have found a college, and you can figure out the financial obstacle course, you are maybe happier staying than moving. This especially if you now have discovered the few financial gems that may exist for you as a student. Some look towards work-study, some are Resident Assistants, some have scholarship help. Others have family.
When you have an established International Student Community and Student Association, it makes a big difference when you need to decide on applying. It also makes a difference when you consider transferring. I think others have hit the nail on the head. A big part is finances but the cultural environment you have established on campus makes decisions very easy for some incoming international students. The colleges that don't have high numbers maybe work with other sister/brother colleges to have students do a study exchange sometimes. International recruiting is big. Lets face it, students are also looking towards Europe, Australia and Asia is creating a lot of incentives to attract future students (China and South Korea come to mind)
It's a lot of pieces that either fall into place or not to give a student a sense of hope. Some get here, mess up and then drop out but that's a different conversation altogether.