Hello Alex
The manual for your VFD should show how to connect a remote start switch. As you have said, the switch only controls the operation of the VFD. It does not switch the input power to the VFD nor does it switch the power going to the motor. Only a tiny amount of power, at low voltage, would run through the switch. You have made a good choice. That footswitch is rated for harsh environments. You don't want metal dust getting into the footswitch and causing it to make your motor turn on when you did not want it to turn on.
Since your footswitch is actually rated to AC power, the third wire is probably a ground for safety (green wire). The ground wire should be connected to the ground terminal of your VFD.
There are usually several different switch connection schemes shown in the manuals. Chose one where there is a "Run/Stop" switch shown (closed for Run, open for Stop). Connect your switch contacts where shown in the manual. Often there is also a "Forward/Reverse" switch. You can connect a toggle switch, if you need this function. If you only want the machine to run in one direction, then install a jumper (or no jumper) instead of a second switch.
If you are having trouble figuring it out, let me know what the brand and model number of your VFD is. I will see what I can do to help you.
Phil