Storytelling is as old as language itself, and stories are inextricably woven into our day to day lives. We tell stories to each other. And there are some professions --novelists, ministers, teachers, actors, songwriters, therapists, filmmakers, for example-- who incorporate storytelling into their professional roles.
The revival of oral storytelling as a performance art form in the United States in the last quarter of the twentieth century resurrected that ancient breed of storyteller who made a living by telling stories to live audiences. A storytelling festival circuit sprang up as part of the revival, and provided venues for these performers, i.e. "platforms" where they could be heard. In some cases, notably the National Storytelling Festival in Jonesborough, Tennessee, the platforms were literal: performances were held outdoors in tents, so small wooden platforms served as stages, so that the storytellers could be seen. In the early days of the Festival, performers would also perform in the high school gym, or in front of or own top of a wagon with the audience seated on hay bales, but neither the term "gymnasium storytelling" nor "wagon storytelling" stuck.
Platform storytelling, then, is the type of storytelling you'll hear at a storytelling festival, as opposed to the type you hear on a front porch, around a campfire, or the water cooler. You might hear the exact same stories in any of those settings, but what distinguishes platform storytelling is the presence of three key elements: 1) amplified sound (a microphone); 2) a paying audience; and 3) a paid storyteller.
As a phrase, platform storytelling is also used to distinguish it from applied storytelling, which focuses on the use of storytelling within a professional field, such as business, education, mental health, and health care. Although similar to solo theatre, "theatrical storytelling" was never considered an appropriate term, as a significant number of professional storytellers have no theatre training, and many prefer to keep their art form distinct.
The term is essentially jargon, used only among the small academic community that studies storytelling, and only among a small set of storytellers. Willy Claflin was a professional storyteller for fifteen years and never heard the phrase until being invited to perform on the festival circuit. He was stumped the first time he heard the term, because of its similarity to "platform diving." To him, the phrase conjured an image of a circus act with a tall ladder and a diving board.