Current can only flow in loops. This is true for signals as well as for power supply current. Unfortunately, a current loop will emit noise, and the larger the loop, the larger the noise. Noise increases with current and with frequency. A large loop is more likely to receive noise. Loops should therefore be kept as small as possible. This means that every line that may emit or receive noise should have a return path to ground as close to the line as possible.
The best way to make sure that every noisy track has such a return path is to add a complete ground plane to the board. Then the area of the loop will only be the length of the track times the distance between the track and the ground plane. This area is usually much smaller than what can be achieved by routing ground paths, so the noise from a board with a ground plane is therefore much less than the noise from a board without a ground plane.