I'm not familiar with this particular control but it seems straight forward to me. The limit control on the left is your high limit and is connected to your spill switch. This control is low voltage (24Vac) and is connected to the low limit switch on the right. The limit switch on the right closes when the water temp is below a certain setting on the dial above. This keeps the circulating pump from moving cold water thru your lines. Its relay is energized by the control on the left and simply closes the switch connecting the two terminal screws. The wire labeled circulator is line voltage so connect the two wires to the low limit control on the right..it is simply a switch loop to the circulating motor.
The high limit on the left is essentially connected as your spill switch, which is thermally held closed. It is powered by the 24 volt transformer. It's job is to energize the gas valve and start the ignition sequence for the gas burner and open the circuit on high limit. The coils in both limit controls are 24 volt. On a call for heat the high limit closes and starts the burner and the water heats. Once the low limit is reached the circulator switch (one on the right) closes and starts the circulator motor and then the system simply cycles as needed.
I do not believe the whites of the gas wire and thermostat wire connect to the high limit switch but instead are connected together in a wire nut. This is the return to the transformer. The black from the thermostat connects to one screw terminal on the left hand high limit switch and the black of the gas wire connects to the other screw terminal. These limit controls are switches that simply break the hot wires to what they control.