Webbing is a common staple of older furniture construction. It was used as the base for many types of seating: dining room chairs, arm chairs, and sofas. Webbing must be stretched tight in order to remain in place for years without sagging.
Web stretchers are an absolute must for this job. They allow you to stretch the webbing straps as tight as a drum, literally. If you lightly tap properly stretched webbing, it will sound like a drum.