Radio license required?

When I first learned the memory aid for required documents, it was "ARROW".

When I inquired why sometimes I saw "AROW", I remember being told that the
second "R" was dropped because while flying within the USA you don't need
the "Radio License".

I glanced through the FAR/AIM and couldn't quickly put my finger on this, so
I'm putting it out he what's the deal on this? I have seen what may be
conflicting indications about this -- for instance, in the thread about ramp
checks, etc.

When do you need the "Radio License" document, and who cares about it (e.g.,
is the FAA or the FCC)?