Normally a hunter Pro-C would have a transformer in the lower left corner of the enclosure. Yours does not have one, and is getting the 24v power from some other source. You’ll need to trace the black wire on the left side of the enclosure to see where that goes.
This is an ugly wiring job, but you can see where the 120v power comes into that covered area in the lower left, and above that is the transformer to convert to 24v for the controller.
I can’t tell if this is an indoor or outdoor installation, but if outdoor, you will need a Rachio outdoor enclosure, which will have a hard wired plug for the Rachio power brick to plug in to. Otherwise, the zone wiring is pretty straight forward. Like @Thomas_Lerman said, I’d leave the sensor wires unplugged for now, get the Rachio up and running, and decide if you actually want/need it.