A guess regarding the red bulbs:
A properly functioning OT should be a somewhat low resistance from the output tube plate pin to the OT center-tap. Due to the use of the plugs for both the output tubes and the rectifier, you have access to these two points at the tester.
If the bulbs were somewhat high voltage neon, then they would be off when the OT winding has continuity, and light up only when there is a significant voltage difference between the winding end and center-tap, such as when the winding is open. I guess the bulbs could be incandescent, but they would have to be high-voltage types to keep the filament from popping with an open winding.
And you use one bulb per OT half-primary.
This tester is a mystery to me. It looks quite well built, with quality parts and construction. Someone had to spend some money (and time) to make this. That only makes sense if you have to test a LOT of amps for OT faults. But if you're mainly finding out if there's an open, why can't you just stick your meter probes into the output tube plate and rectifier cathode socket pins? So that makes me wonder about a production situation, where you might not have time to pull the amp out of the chassis. If so, why not test the OT before the chassis goes into the cab, or before the OT is installed on the chassis?
It seems like maybe someone got carried away building the ultimate tester that they don't need.