Replace Potentiometer With Arduino Controlled Circuit
You can't do this easily and should not attempt it. You could easily hurt yourself or damage your computer. 

You need a power controller with a DC voltage input that is isolated from the mains. It's far easier to design something like that from scratch than to try to retrofit to a non-isolated phase control. That pot, for example, will be subject to hundreds of volts in operation, far beyond what any digital pot can withstand. 

If you connect your Arduino directly to a non-isolated circuit there's a good chance it will be sitting at 120VAC relative to ground and could electrocute you or blow your PC's motherboard to smithereens via the USB programming port.