It’s a holdover from the early days of cable/satellite TV.

Ted Turner owned the TV station that eventually became TBS, and the Braves. His station aired a lot of Braves games, and that station was widely available on satellite and cable across the country in the late 70s and early 80s.

So for a good number of years, the Braves had the most games on TV nationally, and that’s when they got that nickname.