After registering your rig on the website, you will be asked to add a pool to your miner software, for example:

cgminer --scrypt -o http://eu.betarigs.com:3333 -u -p x -o -u -p ...

That will put the Betarigs "pool" in first priority position, and your usual pool in second position. So:

    Your rig is not rented: The Betarigs "pool" is inactive and your miner mine your usual pool.
    Your rig is rented: The Betarigs "pool" becomes active and your miner mine it. Behind the scenes, Betarigs will redirect the miner to the pool chosen by the renter.