What kind of magic is energizing the coils in this simple stepper motor driver?
The schematic seems to have an error.

The center taps shouldn't just be connected to each other, they should also connect to the +12 V supply.

On your linked page, this is confirmed in the text:


  The Stepper Motor is a Unipolar Type in 5 wire configuration. The center pin is shorted internally and is connected to the supply (12V here).


And in another diagram on that page:



And on their breadboard: