What is the purpose of this “red room” in “Stranger Things”?

It's a dark room for exposing film to photographs, from way back when you did it by hand using pans of chemicals. The light allows you to see enough to work without exposing the photo sensitive materials and ruining your work.

"When making black-and-white prints, a safelight is commonly used to illuminate the work area. Since the majority of black-and-white papers are sensitive to only blue, or to blue and green light, a red- or amber-colored light can be safely used without exposing the paper" - https://en.wikipedia.org/wiki/Darkroom