Red and black are the two electrodes inside the strobe tube. Red is the anode (+) and black is the cathode (-). These apply a voltage of about 500 V to the xenon gas inside the tube, which is normally non-conductive.
White is the trigger, which is the wire or band wrapped around the outside of the tube. When the strobe fires, the trigger briefly applies 7,000 V to the tube, which causes the xenon gas to ionize and become conductive. Current flows though the gas from anode to cathode, producing an intense flash (basically a lightning strike in a tube). The gas goes back to the non-conductive state within a few microseconds, which is the flash is so brief.