I've noticed that this expression is familiarly used to indicate something being in style, cool, or generally "à la mode" - notably here on this stack exchange site.
Does anyone know the origin of this usage ?
This is related to brancher as in :
brancher une conduite (une canalisation) à un circuit principal.
voir branchement électrique.
Branché recently received (particularly among youngsters) the meaning of:
mis au courant, concerné (1973)
and took by extension the meaning of :
à la mode, dans le coup (1980).
Cablé (1980) as the same meaning.
Source: Dictionnaire historique de la langue française, Alain Rey. Approximate translation.
Those of us who are more than forty may remember the interview of F. Mitterrand by Y. Mourousi ;) see the video from INA
I would say that it is something derivated from the avoir de la branche expression :
So the origin would come from the prestige of being part of a noble family tree (Arbre généalogique).