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How does this work?

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It works on the voltage drop across R7 - however, you really don't want to be using crude linear regulators for a solar powered project - you're going to wasting the biggest part of your collected energy (probably 80%+ with the voltages and currents you're talking about). You really need a much more complicated (but vastly more efficient) switch-mode supply.
 
Either use less cells in series and a linear regulator or use a switching regulator.

Some of the ICs around are pretty simple to use, all you do is ad an inductor, a couple of resistors and a MOSFET.

Are you sure your panel will give 15A in the ambient lighting conditions? It might be rated at 15A but that's probably only at high noon on the equator with full sun. If you live in a cloudy place nearer to either pole then you sould divide the power output by a factor of four or more.
 
The original circuit uses 0.1 ohms resistors at the emitters of the power transistors. Your 1k resistors are 10000 times higher in value so the transistors will do nothing.

I manipulated your awful black background and cropped your schematic so it can be easily seen.
 

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