I will try to explain in the simplest way I can.
Imagine an audio power amplifier.
The input signal is very low (from zero to about 1 V rms), and the current flowing from the power supply is high (in the order of some ampere)
This current is DC, but it may contain a lot of noise, due to the AC ripple and to the current supplied to the amplifier (current that can be very large).
We have also the output ground terminal, that carry a large current (that "sorts" from the amplifier and "returns" to the amplifier ground).
These large currents pass on the PCB tracks (or on the wires, in case of a point-to-point construction) and are transformed in voltages over the tracks (or wires) impedances.
These voltages are very small, but in the order of the input voltage. Depending on where the input, power and output grounds are connected, and how they are connected together, it is possible that these unwanted voltages add with the input signal, resulting in noise and sometimes distortion or instability.
One way of wiring that helps in reducing these effects is "star grounding": all ground points (input, output, power supply) are carried separately to a single point (our star center) that is the main ground point.
In this way there is no mixing of currents or associated voltages.
In audio amplifiers this type of wiring is the one that usually results in better performance.
Refer to the attached images, for a schematic (look at the ticker traces) and a PCB.
Most of my "older" amplifiers were designed with single sided PCBs.
There was no ground plane, but their performances were similar to the new ones, that are mostly on dual sided, ground planed PCB.
My experience is mixed: sometimes the ground plane helps, sometimes results in lower performance, especially regarding noise.
For digital circuits, and mixed analog-digital, the ground plane is mandatory, but it is helpful, during component placement, to follow the old-style rules, and place ground tracks in "star" style.
Later these ground tracks may be deleted, and the current will flow in the ground plane following these "ideal" paths. This will be of no help for EMI issues, but will increase noise and distortion performance.