When paper was widely used in schools and offices, it was common to fold the paper into various shapes for fun, or for concealing the contents of a note. Paper airplanes were generally just for fun, but could be used to convey a message, sometimes a surprisingly long distance (if thrown from the roof, for instance).
The information-carrying capacity of paper combines with the forward, directed movement and distance of a paper airplane to make a simple, reliable image that’s reminiscent of an arrow but unmistakably associated with documents and text.
The form of the paper airplane has become so stylized that it is now almost interchangeable with the arrow, and often the two are combined in a sort of split-arrow/2D-plane hybrid.