The term "low key" means to do something subtly. In the example that you gave you used bed wetting. Bed wetting can be embarrassing for a child to deal with so approaching the situation with a low key attitude means that you are going to approach the subject gently and subtly. This term can be used to describe any situation that you want to approach without making a big scene.

The origin may have been a technical term in photography and cinematography. The key light is the main light. "Low-key lighting" means that the scene has a dark look, overall. By extension, "low-key" means any situation that is gentle, relaxed, quiet, not dramatic.