Hi there Ron,
In theory this is how it works...
When power is first applied, C2 should have around 4v across it. That's because the two diodes are Schottky diodes not regular Si diodes.
With 0v across C3 the output of the LM358 is high, which means it might have 3v at the output or maybe a bit less, so C1 charges up just a little bit with left side negative and right side positive. C3 starts to charge, and eventually becomes higher than the 2 x 100k voltage divider voltage (note an improvement might be to power the voltage divider from C2 rather than from +5v). The output of the LM358 ramps down low toward 0 volts. As the output gets lower, C1 charges up more with the most positive still on the right side. Meanwhile, C3 is discharging. As C3 voltage gets lower, the output of the LM358 again goes high, forcing the left side of C1 higher than before. Since before C1 left side was near 0 volts and the right side was near 4.5v, the cap voltage is around 4 volts. With the left side near 4v (the voltage across C2) and the cap voltage around 4v, the right side then goes to around 8v. This pumps up C2.
This was the quick view and in real life it should take some cycles to pump C2 up to it's maximum.
Remember this circuit has not yet been tried out in real life. It should work because C1 charges up based on the upper diode and the LM358 ground terminal, so C1 charging does not depend on the voltage present on the positive power supply pin of the LM358. Thus the energy entering the pump part of the circuit comes from C1 charging up.
Also, when C1 discharges it is partially floating so the current should divide between C2 and the positive supply pin of the LM358.
Does it really work in real life? We'd have to try it and see. If it does in fact work there may be an idea frequency to run it at as well as ideal cap values for C1 and C2. I started with the ratio 5:1 but that was just a quick guess.