The backflow preventer means that there is no place for the pressure to go as the water heats and expands. The anti-back-flow device would be installed right at the pressure regulator coming in from the city water. The city is concerned about ground water (from the sprinklers, remote but possible situation) getting back into the system, especially when pressure is lost in the public main, and then contaminating the main and other peoples water. By using the anti-back-flow device this creates a closed system. The only pressure relief is the TPR valve on the water heater. A Bladder tank as illustrated above, or if you are on a well, you already had a means of expansion in the well water bladder tank, is a very safe way of allowing for expansion. When you install a sprinkler system, you most likely need a permit, and this creates an opportunity for the city to force you to install a Anti-back-flow device (and probably a regulator also). Otherwise they are not in a good situation to require every house to install an anti-back-flow device.
There was a point in time in the past where there was a toilet valve that was improperly designed and allowed water from the tank (think blue water from a bottle of Vanish) to get back into the public water system, during a failure of the water pressure (busted main, repairs necessitating a shut off of the water, etc) and this is one of the big reasons that anti-back-flow devices became a requirement. Its mandatory most places now, and should be.