Those screens-within-a-screen are called waipu in Japanese, which would seem to come from the English word “wipe.” As you may know if you’re a film buff or work in television, “wipe” is a film-editing term. It describes a type of film transition where one shot replaces another by moving from one side of the frame to the other, or with a growing or shrinking shape. (Think of the shrinking circle used in the old “Looney Tunes” cartoons to mark the end of the story.)
Now wipe those images from your mind, because what we have here is another case of an English word borrowed into Japanese to mean something else. A waipu, which is what you’re asking about, is something different. It is a superimposed image, smaller than the main frame and usually in the shape of a box, which is used (and overused) in certain Japanese television programs to show the reaction of a celebrity guest to a video clip.