Rear lights go dark when brakes are applied... why?

More than likely. :eek:
I make it a point to try and get the drivers attention (if I can do it safely) when I happen upon someone with no rear lights, and let them know that their tail/brake lights aren’t working.
And more often than not, they’re unaware that they had a problem.

Are you sure about this? I’ve owned several (older mostly) cars that had separate filaments for running lights and brake lights. The running light filament goes out when the brake light comes on.

If you mean that a lit taillight goes out but the brake light does not come on when the brake light is energized, that’s the symptom described in the OP. If you mean a lit taillight goes out when the brake light comes on, I submit you are mistaken, with the much brighter brake light obscuring the comparitively dim taillight.

Regardless, yes I am sure that a burnt out brake light filament cannot cause the symtpom described in the OP. It will prevent the brake light from coming on, but will not affect the taillight one way or the other.