What is the gold-coloured area on this circuit board and what is it for?

The gold colour is just the surface finish of the PCB, called Gold Flashing.

The exposed areas could have a couple of uses:

• As a grounding/earthing point, for example an RF screening gasket or can might make contact there.
• Heat-sinking for a device. We have very similar footprints on boards for Allegro A4988 stepper motor driver chips and switch-mode power supply chips, but they can be anything that requires heat sinking.

There may be other reasons, those are the two I'm most familiar with.

Edit: Another reason is to do with RF performance or when RF devices such as filters or antennas are made purely using tracks on the PCB, in those cases the solder mask (and every other layer of the board) has an effect on the fabricated device, but I'm not an RF engineer so can't really expand on that other than being something I've seen in equipment we've designed.

It looks like the exposed copper is gold plated. e.g. Wikipedia Electroless nickel immersion gold

That is a very common for SMD PCBs because it gives a lovely flat surface to solder SMD to.

Unlike other finishes, gold plating doesn't tarnish, or oxidise, so the PCB can be stored for months without it deteriorating.