What sort of crazy ignition gimmick is this?

67Dart, thanks for the clarification and details on the CDI system. My only question at this point is the red button on the CDI box. Someone mentioned having it pushed in or out enables / disables the CDI, and leaves it to run on the stock system, but I'm not sure which is which.

I went and got some gas today (ethanol free, I wanted to start the car off right), and tomorrow or the next day, I will hook up a 1 gallon gas can to the fuel pump, crank it, and get the carburetor filled with fuel. Then reinstall the spark plugs and see what happens. I don't plan to run it for long, just to see if there's a running engine there.