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