I work in alumni relations at my old boarding school and give regularly to my alma matter, I'm 25. It is shocking to see how many people don't see how philanthropy benefits these types of institutions. It is because of the donations of alumni, family, friends, foundations, etc. that these places are able to function.
Many times the operating budgets are dictated by how much money the annual fund makes per year. This effects everything from: financial aid for deserving students who make the institution stronger; faculty development that keeps strong and dedicated faculty at the school; facility maintenance; support for the arts, athletics, and student programming etc. etc.
Additionally, the "well I payed tuition argument" doesn't hold any water. If you look at the operating budgets for these schools and divide by the number of students, you get the what it actual costs to support a student at the school. This is often much, much lower than what you pay in tuition.
Obviously, no one is obligated to give back. But it helps support the students that were there. Just like you benefited from the generosity of others while you were there.