A high-frequency or fast rise-time pulse requires a ground plane in close proximity to carry the return signal. else the signal will radiate and the high frequency signal or signal edge becomes distorted. Even a small gap in the ground plane which causes the return signal to take a different path can cause significant problems with the signal. For very high frequencies, microstrip traces are used where the trace width, and distance to the ground plane are controlled to maintain a specific impedance and minimize signal reflections.
A low frequency return signal can take a longer or different path without serious radiation or distortion so a ground plane is not as essential. The frequency at which a ground plane is required is not exact. Generally to be safe, a ground plane is used for most circuit boards operating much above audio frequencies including most digital boards.