The payphone was popular back when no one had cell phones to make calls while on the road. It's really just a public telephone that requires coins (another soon-to-be retronym?) in order to be used. The term came about in the 1930s.
Of course, nowadays most people have their own phones to use whenever needed, such as while traveling or when going out to the store. This makes calling someone to alert them about your ETA, your purchases, your car breaking down much easier—and takes the hassle out of digging out spare change.