Increasing the plating thickness of copper does increase its ability to remove heat. Increasing the number of copper planes also works, as long as they are stitched together by an adequate number of via holes, whose copper sides will conduct heat between the planes.
As George pointed out, extra copper area also improves heatsinking. To get the full benefit of copper plane heatsinking, you need both area and thickness. A small thick area round your device isn't able to lose much heat, although it conducts it well. A large thin area isn't able to pull much heat out of the device, although it can lose it well. You need both thickness, and potentially multiple planes around the device to minimise the temperature drop to the rest of the heatsink, then a large area to lose the heat without too much temperature rise above ambient.
If you are to do any serious board designing for heat, then you need a thermometer, and to measure the temperature of different parts of your prototype when it's dissipating power. If your copper plane is hot, it needs more area, or better connection to chassis or heatsink to lose the heat. If your copper plane is cool and your device hot, then you need more copper thickness around the device.
With enough via holes, the heatsink can be placed on the bottom of the board, and remove heat from the component on the top side. A fully filled via, whether fully plated, or solder-filled, conducts heat better than a typical tubular via.
Be warned though that increasing the amount of copper round and/or under a part's footprint makes rework increasingly difficult, or increasingly reliant on devices with better control and more heat delivery capacity than a simple soldering iron.