1) It puts you in a network of other people who went to the institution, oftentimes people who have connections that can help you in life
2) It gives you a piece of paper that shows you can follow a long term plan and execute on it, that other people recognize as a difficult long term plan to execute.
3) Gives you an icebreaker when going to interviews where you can talk about either how you and the interviewer went to the same institution, or rival institutions, or you can talk about how your elite institution was different from theirs and the same.
What you don't get is a "better education". (Edit: Some have pointed out rightly that I should differentiate here between top non-elite instututions and the ones that are diploma mills, so to be clear, what I'm saying doesn't apply to the diploma mills.) You do the same work as the folks at the less elite institutions. Maybe you do it with slightly smarter people who help you learn new things that you wouldn't otherwise, but most of the "less elite" schools still use the same textbooks and materials. And heck maybe your professor wrote the book so you can get more detailed answers, but it's unlikely that that access gives you a significant leg up unless you take advantage of it.
So it makes sense that in fields where skills matter, the elite college doesn't add a lot of benefit, but where connections matter, it does.