When I say click off, it doesn't spin continuously like an encoder. It has off/minimum level, and max level...like a volume.
Sometimes pots have a switch at the end of their travel, so if you carry on turning anticlockwise you will feel it click. However, I have only ever found these with a LOG pot, sometimes called an analog taper.
The Arduino can do what that board did and generate a PWM signal. See this for how PWM dimming works http://www.thebox.myzen.co.uk/Tutorial/PWM.html.
The only difference is that the board on eBay will handle more voltage and current than a direct Arduino pin. To handle the same voltage and current on an Arduino you have to add a logic level FET. One convenient way of doing this is this product although their are many more. Arduino FET Shield