It's called "ground fill". It allows for easier ground connections, and can also be cheaper to produce. In the routing process it makes it easier to connect pads because there won't have to be ground traces all over the place--just a solid plane on the board. There are quite a few advantages to it.
The ground plane also suppresses radiated energy a little bit, if there is a copper plane on the other side of the board it creates a bit of capacitance to smooth the power supply, and it takes less chemicals to not etch away all that copper that isn't hurting anything.
There are probably more advantages that we haven't thought of yet.
Think about this: the current from every component on the board has to go through the ground wire. If you have too much resistance, all those currents cause the gound voltage to bouce up and down. When that happens, every component that depends on ground for 0V reference is instead getting a voltage that is jumping, and thus isn't operating as they should.