This video is far more relevant than it might first appear. If you understand this video properly, and then go back to the schematic you posted, it'll make sense:
Sure. the video is a doubler rather than the negative rail that you want, but just flip the diodes around the other way and you'll get your negative rail instead. And the circuit in this thread uses AC as the oscillating signal, rather than a microcontroller pin. Same principle.
If you're wondering why a doubler looks so similar to an inverter, you just have to realise that a doubler and an inverter are exactly the same thing. Think about it: you have 12V (or whatever) and 0V, and you want to generate -12V. Now, measure those voltages relative to the top rail -- the top rail is 0V, the old ground is -12V, and you want -24V. You want a voltage doubler, albeit one that operates on negative rails. Just flip those diodes around.