What is the purpose of copper planes in a switching power supply?
Lower track impedance

In a switching regulator, the track impedance matters a lot. Not only resistance, but also inductance, and both are reduced when using wider tracks (or planes).

Heatsinking

A switching regulator produces heat, which has to be channeled out of the component. Copper is a very good heat conductor and is used as radiator in many switched mode power supply designs.

PCB Manufacturing issues

When producing PCBs, manufacturers often ask for a certain percentage of each layer to be copper. This is to ensure an even thickness on the whole PCB in the plating phase, as well as an uniform expanding and shrinking under temperature variations.