First you need to identify the controller lines and solder 2 wires from the switch

I drilled a hole through the top of the controller and sent the 2 lines out from the opening

When all is done it should look somehting like this. The fog machine requires a little bit to warm up and thats why they have a momentary switch.

Now put the relay shield on and upload the code, then wire it up. My setup I put the bluetooth module to the RX TX of the Arduino and the RF to pin 2

The Android app has 2 buttons turn the fog machine on for the time on the button labels. I implemented a countdown timer so you know when you can send another command.