Can I get a layman's understanding of what is going on here?

[Theckie]

As circled, what is all this?
Is there a better way to achieve the same function?

I can tell there is a 2-way Solenoid Valve (NC 24VAC 5-1250 PSI 1/4" NPT) and a 3 way valve.

Thanks

 

[MEP001]

That appears to be a system to run the spot-free rinse through the high pressure pump at a reduced pressure. The unit to the right in the circle is an Erie Motortrol valve that switches the water source between city water and spot-free. The solenoid opens when spot-free is selected to allow the second regulator to drop the pressure.

The "better way" is to have a separate pump and solenoids for spot-free and tee the line in at the boom. Your changeover time might be a minute or more now, but with it tee'd in at the boom it will be five seconds at most if you run 250 PSI.