You are correct, the phrase dates back to the days of early radio (when it was still referred to as the "wireless.") It often took some delicate maneuvering with the radio dial to receive a station clearly, without static or background noise. This procedure was referred to as "tuning in." If you were to switch stations, it took a good few minutes to tune into the new station properly. So to discourage listeners from changing the station, radio announcers would advise them to "stay tuned" and back up their plea by promising interesting stories and programming after the station break.
In the early days of television, TV sets had a set of "rabbit ears" antennae on top, and whenever you chose a station, you had to shift the antennae this way and that to get optimum reception. Likewise, you had to adjust the "fine tuning" dial, as well as the horizontal and vertical hold. Today, of course, most televisions have automatic, electronic tuning and all you have to do to change a channel is push a button and you get a perfect picture every time. But the phrase "stay tuned" has remained in the broadcast vernacular.
(Likewise, some announcers will still warn viewers "don't touch that dial!" meaning don't change the channel. This dates back to when TV sets had rotary dials that were used to change channels. Dials have since gone the way of the dinosaur, but sometimes that phrase is still used.)