When this circuit is all up and running the two solenoids open up.

Can someone please explain to me how this circuit works?

When the arming switch is on it joins pins...
When the microswitch is activated...

nitrous setup ;)
when the switch is on it allows power to the microswitch (which would be on the accelerator, so only triggered on WOT). this allows power to trigger the relay which in turn powers the solinoids.

so if I wanted to use a pushbutton instead of the microswitch would I just remove the microswitch and put in a push button inside the cabin?

in essence yes... but if it is for nitrous ur better off having the wot pedal switch... much safer

after key is on turn the arming switch on, then when the micro switch is turned on the relay turns on turning the solenoids on. letting nos flow