If you're part of the tiny elite for whom Harvard and Stanford are an option, by all means go for it. But for a lot of people, studying abroad might be a better idea.
1) The professors. Imagine taking classes and interacting with Nobel prize winners, or other famous people in your field (let's say, taking a class with Knuth at Stanford).
2) The networking/social aspects. Imagine being college roommates with "the next Mark Zuckerberg", whoever that may be.
3) More cynically, the pedigree. This one isn't strictly required, but it helps enormously if you want to get into academia, and it can also help you with an industry career or start a startup (it can make things easier).