alt Hacker NewsRescue operations, etc...
I saw a presentation by one of the founders where he talked about several use cases where the benefit is just phenomenal.
They don't fool me for a second, however. The end goal of this is to build a weapon that can fry people/places on demand (but only the bad guys, of course).
orbital mirrors make very poor weapons because Conservation of Etendue means you fundamentally cannot concentrate very high unless your fill factor is very high (which would require millions of tons in orbit). Lasers are far more effective as weapons as a single aperture launched on a single rocket is sufficient to get high concentration. Microwaves also.
The military application would just be illumination.