The "voltage" that is used in electronics is always a difference in potential. GND is just a common reference, such that Vcc-GND = say, 5v. Since we always measure a difference, then GND must be defined as the 0v point, since GND-GND = 0v. Thus, any connection to 0v is a connection to GND.
When the controller mentions HIGH and LOW, it really means voltage of Vcc and voltage of GND, respectively. Since all grounds should be common within a circuit, and so all voltages reference to GND, we just say 5v and 0v to make things simple.