In that context (sounds like a book?) it probably means a dance. College's in America don't really have dances, unless one is in a Fraternity or Sorority. And they wouldn't call them flings. The reference sounds like it's from the 50's honestly. We always just called them dances. Although I have heard of spring flings. Usually a Fling is a reference to a short romantic relationship. Such as a "summer fling" would mean you had a relationship with someone over the summer, but for whatever reason (usually one outside of your control) it ended.

I've actually googled this, and it seems that a fling (more specifically, a spring fling) might mean a college dance in some American universities (according to urban dictionary). Here's the link: http://www.urbandictionary.com/define.php?term=Spring%20Fling It makes sense in the context of your text. But you'd better get some answers from native speakers who have actually heard about this thing. They'd be more reliable than urban dictionary :D