How would I use an arduino to detect the state of a dimmer switch?
You could remove the circuitry and wire directly to the pot in the dimmer but typically the pot in a mains dimmer is too high value to work directly with an Arduino ADC input- you would normally want something around 10K and the dimmer pot is usually much higher value. 

It's possible to use a rail-to-rail op-amp such as a MCP6002 to buffer the pot wiper voltage, but I think it would be easier to just buy a pot and knob.