When water freezes in a pipe and the faucet is closed the water between the faucet and the frozen part cannot escape. This leads to increase in pressure as more water freezes. Because as long as there is still unfrozen water in the pipe the ice expands along the pipe. The increase in pressure is what causes a part of the pipe between the frozen part and the faucet to burst.
Now if we let the faucet trickle and assume the ice freezes in the same spot as when we do not let the faucet trickle the ice expanding from the frozen part will not be able to lead to a build up of pressure.
So apart from the fact that water freezes less likely when it is moving letting the faucet trickle should mainly be done to release the pressure between the faucet and the frozen part of the pipe. Note that ice can always expand in the other direction (to the water supply side) by displacing water.
Is this mechanism correct?
EDIT: Found this interesting video regarding pressure buildup
EDIT2 : After further reading, I came across the term latent heat. Because the evaporation of water cools the surrounding environment letting a faucet drip can actually accelerate freezing. But the freezing will then probably occurs first at the faucet itself. Given the above considerations, this could actually be beneficial in as such that there is no water trapped because the freezing starts at the faucet and gradually moves upstreams.