The gui probably just has a simple serial interface set at a high bitrate (19200 would probably be enough bandwidth for this.) and a matrix (like any other led matrix) that controls relays rather then leds. Also, by the looks of things, it didn't support using rows, only columns, so it probably didn't even have a matrix, just relays for the entire column.
Agreed. It's a simple serial monitor thing. If its the arduino running the entire thing, then he's just got it plugged in to his computer and then is using a serial monitor. There are serial monitor examples in the file menu of the arduino programming console.
And then the program that is running the whole thing looks like it was programmed separately to interpret the serial sending from the arduino.
The program he wrote to interpret it looks fairly basic.