I hate statements like this due to their imprecision and their contribution to making mathematics difficult to learn.

> Though they’re defined by just a few rules, groups help illuminate an astonishing range of mysteries.

An astute reader at this point will go look up the definition of groups and come away completely mystified how they illuminate anything (hint: they do not).

A better statement is that many things that illuminate a wide range of mysteries form groups. By themselves, the group laws regarding these things tell you very little. It's the various individual or collective behaviors of certain groups that illuminate these areas.