Red background means a job that's more dangerous than usual (the danger is real, main characters are at risk, can be seriously harmed or even die). Most of the stories are about jobs given by one temp agency, the red ones are given by another, more sinister agency that Nagomi has to work with in order to find out what happened to her dad. The use of red in these comics tends to tell us "this is the threat". One time, the mangaka tried to switch it up by having the danger color be green, but it seems like they went back to red—as well they should! White/black/red is a true and tested color scheme.
I really have to commend the mangaka yet again on being willing to play with format. Relying on the buildup of episodic practice, of tropes around style—the red color's appearance instantly tells us what the stakes are, we don't even have to get an explanation of what the job is before we're already on board. I also love the increasing usage of "regular people" as the beginning POV, only introducing the main characters midway through the first chapter. It's a little bit Whovian, isn't it? the Doctor can't die, but random schmucks certainly can. So make us care about the schmucks, and the stakes are instantly raised.
