One set of <Angle Brackets> is used for "real" bindings - ones that refer to actual mouse or keyboard actions.
When <<Two Sets>> are used, it's for a "virtual" event binding, where you have a binding with a custom name and can then assign any number of "real" bindings to trigger it with the [event] command. (For instance, the [text] widget binds to the virtual <<Copy>> and <<Paste>> events, and these are triggered by <Control-c> & <Insert>, and <Control-v> & <Shift-Insert>, respectively.)