I would say that it is something derivated from the avoir de la branche expression :

1. Avoir de l’allure, de la distinction.
2. (Généalogie) Appartenir à la noblesse, avoir des ancêtres issus de
   la noblesse.

So the origin would come from the prestige of being part of a noble family tree (Arbre généalogique).

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