OK, If you use the control and foot switch as built:
1) The foot switch controls an SSR...which is nothing but a relay. The SSR can switch any voltage. It doesn't care if it is 24VDC or 240VAC. It just turns the circuit on or off. Whatever voltage is running through the run/stop switch is going to be turned on and off. Most likely it is 5-10VDC ( see last paragraph).
2)The control voltage from the foot switch to the SSR is low voltage and is completely isolated from the circuit the SSR controls. ( see #1)
3) Many people add a remote switch to their VFD. The SSR goes where this switch would go ( and the foot switch turn the SSR on and off). In electronic terms, the Run/Stop switch and the SSR load contacts are wired in parallel, and the foot switch energizes the SSR.
The reason for the low voltage wiring to the foot switch is mainly as the Count noted - the switching is done by a device made to switch large loads many thousands of times. The secondary reason, is that if a hot piece of steel or some sharp/heavy object cuts through the cord and shorts the leads to the switch, all that will happen is the device will turn on. There will be no sparks, shock, or danger. In such an event, just pull the plug from the foot switch where it plugs into the control box.
NOTE, It is a bad thing to turn a VFD on and off by turning the power to the VFD on and off. That is why there is a separate switch for Run/stop from the main power. Controlling the 120/240VAC input to the VFD would be the wrong place to install the SSR.
Now, if the voltage through the Run/Stop switch is only a few volts DC, then you don't need the SSR at all. Just wire a foot switch in for the remote switch, and you are in business.