Not always fossils. But their core mission is to teach, and, well, when's the last time you went into a university museum?
The art museums get some foot traffic. But natural history? Last time I went into one (University of Texas, flagship school, large museum) my wife and I were the only people there for the 2 or so hours we were there. Some clerk came out 20 minutes after we entered, seemed surprised that somebody was there, asked if we needed anything, and then vanished. Apparently their primary foot traffic is people who work nearby and need a quiet, sophisticated place to sit and drink their coffee and eat lunch, then come in from the lobby to use the bathroom.
They need to justify their funding, so administrators are okay with it. Plus, having known a curator or two, they really like their fields and would do this kind of stuff anyway. The Boyce Thompson Arboretum in Superior, AZ, for instance (part of the U. Arizona system) had tours and in addition had their curators do field trips. They'd bundle up stuff, exhibits and whatnot, and go into the 'field' of classrooms in the great Phoenix area to teach about wildlife and natural history.
Science and tech museums that don't do this kind of thing and aren't just humongous are called "storerooms", where everything's in boxes, slowly moldering.