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.

Heat and low impedance for the high current paths. Some of the land area on this board might be non-critical but when you have the empty board space it provides a little extra safety margin.

It is generally not a good idea to create a large land area for the switching node (probably on the lower right of this image, but hard to tell without part numbers/schematic/etc) because the fast edges on the switching node can be an EMI problem and the large land area creates both an antenna and capacitively (sp?) couples the signal to the ground plane and possibly other traces which can create ground noise.