I don't know how it works at a university level, but at least when i did some admin stuff for a university department office 365 was free for students and staff.

I don't think the university pays for that, I believe MS just offers them for free to some folks to get them hooked, it's the same as BizSpark and other initiatives.

I can confirm that University pays for MS EES Agreement. In return you can, for example, install office on every pc in Active Directory, upgrade to every windows operating system on pc owned by the University (for example you buy Windows Home on a pc, you can upgrade to professional edition) and so on.

You're right, it's a paid service, but they usually get substantial discounts (smaller schools can even get it for free if they have the right contacts)