The black and red outputs are isolated and floating. They have no relation to earth or mains input.
You use the green earth pin if you want to reference some point of your device to earth. If not then don't use it.
A case when you might want to use the green earth terminal is if you have a device with metal case and you want to make sure the metal shield is not floating but connected to earth. The device might be a sensitive amplifier or a high voltage generator. When earthed, it can reduce surrounding noise being coupled to circuit, or it may add safety if your high voltage generator malfunctions and the case comes live with high voltage.
And there are also cases when you likely don't want to use it. For example if your powered device also connects to another earthed device (like desktop PC over USB or serial connection), if the ground between device and PC malfunctions, the return current will flow via mains earth wiring inside walls.