Letting a faucet drip during extreme cold weather can prevent a pipe from bursting. It’s not that a small flow of water prevents freezing; this helps, but water can freeze even with a slow flow.
Rather, opening a faucet will provide relief from the excessive pressure that builds between the faucet and the ice blockage when freezing occurs. If there is no excessive water pressure, there is no burst pipe, even if the water inside the pipe freezes.
A dripping faucet wastes some water, so only pipes vulnerable to freezing (ones that run through an unheated or unprotected space) should be left with the water flowing. The drip can be very slight. Even the slowest drip at normal pressure will provide pressure relief when needed. Where both hot and cold lines serve a spigot, make sure each one contributes to the drip, since both are subjected to freezing. If the dripping stops, leave the faucet(s) open, since a pipe may have frozen and will still need pressure relief.