The excitation pins are to supply the field winding of a wound field motor. If your motor is a permanent magnet type you can ignore these.

In principle you control this from Mach3 by smoothing the PWM output (unless the card already does that) to get a variable voltage and feeding it to the terminal to which the slider of the pot would be connected to, also connecting the GND connection of the breakout card to the negative terminal of the pot. Speed controller usually use a 0-10v signal and yours would hopefully be similar but the documentation is silent on this.

HOWEVER!!!

A key point is whether the unit is fully isolated from the mains. The photo of the PCB does not show a convincing isolation transformer. The "ground" of the unit may be live when the unit is operating, in which case unless your breakout card has isolation you will get a "loud brown smell" when you connect the mains to the controller and the breakout card and/or the controller may be destroyed. If you're lucky you may survive as long as you're not touching the metalwork.

So it is important to establish if the control voltage input is isolated, or if the breakout card has isolation. If not isolator circuits are available, or are easily built if you are comfortable with electronics.