Answer to OP’s question:
Because elite private schools offer the same kinds of privileges this economic class is used to and expects, that’s why. Example on the K-12 level: One private high school I am well acquainted with has a full staff in their college counseling department. For a senior class of about 150 students, there is one head of cc, three associate heads, one full-time person devoted strictly to assisting with testing accommodations and scholarships, and one admin assistant. Compare that with your typical public high school, where one single director of cc and her/his assistant is assigned to the entire school, and the senior class size ranges from 300-500.
In addition, teacher load accounts for similar differences and can affect the availability of the best teachers to write recommendations for applicants. My public school students find it much more difficult to secure teacher LOR’s for all kinds of purposes (college, summer internships, competitive summer programs, etc.)
It’s not just about “prestige.” Abundant resources in K-12 schools are more likely to ensure a smooth transition to a college with abundant resources. And some of this has much to do with simply daily life for the student. Consider Princeton University, a campus which is so clean and beautifully maintained that it literally looks like a Hollywood set. When you visit, it’s difficult to believe it’s “for real.” Never have I seen a skating rink so utterly pristine. Paradise comes to mind.
Dorms are immaculate. They could be compared with this:
https://alt1053.radio.com/blogs/kcbs-radio/uc-berkeley-stern-hall-and-foothill-hall-are-infested-rats-and-insects-students-say
http://www.ktvu.com/news/berkeley-students-encounter-mice-mold-bed-bugs-in-dorms