This seems much more likely than the other answers given here. The extra heat introduced by a slow drip isn’t going to be significant compared to the exposure of a bunch of copper pipes to below-freezing temperatures, nor does the flowing theory make sense to me. But if you relieve the pressure, the pipes are able to freeze–just without bursting the pipes.
For those of you that have used the dripping water trick in deep freezes: have you ever noticed the rate of flow (with a wide-open faucet) drop after a long freeze? If so, that might imply that the pipe has partially frozen, constricting the flow–but due to the pressure relief, hasn’t damaged the pipe.