What is the purpose of the brown louver-like structure on the underside of a mushroom?
These thin, closely-spaced structures are commonly called gills (in more formal contexts a gill is called a lamella).  Our most familiar mushrooms have gills, but there are other species that do not have structures of this form.

Photo by Jason Hollinger via WikiMedia.org

The gills exist for reproduction. Mushrooms spread by producing tiny spores that disperse on air currents.  (In fact the entire mushroom structure exists for this purpose; the main part of the organism, a fungus, is a diffuse web under the ground or other substrate.)

Mushroom spores develop on surface tissues so that they can be released into the air.  The gills evolved to provide more surface area so that more spores can be produced and released (up to twenty times the area of the supporting surface).  The gill spacing is important because the spores are shot out of the surface, are slowed by air resistance, and then fall downwards from between the gills. (Source: Why mushrooms form gills: efficiency of the lamellate morphology)