Okay, the abstract clearly had english words in there, but I've got no idea what they mean. Does anyone have an overview that would make sense to a non-expert?
A group is a collection of objects that you can transform via certain defined methods and they respond in known ways.
A metaplectic group is like a mirrored version of a group you already know, with a few other changes in the behavior. Internally similar enough that in seeking to understand it you’re not starting from scratch; the group feels familiar.