The simple answer is that a sufficient amount of flowing water above freezing temperature through a cold section of pipe adds enough heat in that area relative to the cooling rate outside the pipe to keep water in that section from freezing completely through, which would allow potential bursting via the expansion of freezing water in a closed system.
Flowing water does freeze; it freezes when sufficient energy/heat is removed, and can do so in a few forms not everyone has seen. Most people are familiar with surface ice forming on rivers when the atmosphere gets cold enough. You can also have conditions that allow frazil ice to form, which are patches of slush that form and keep moving with the flowing water. Anchor ice can form on the bottom of a stream, even when the surface is not frozen. An aerator keeps a small section of surface ice open by creating a slow steady flow of deep (relatively) warm water at 0-4C up to the surface and prevents or limits the cold -20C atmosphere from forming surface ice.
Once water starts freezing in a pipe or any other partially or fully enclosed area, it may or may not stick/expand/burst the confinement based on the shape of the vessel, how smooth the ice-wall interface is, how fast the ice forms, and how strong the confining material is. That’s why sometimes a cheap plastic outdoor spigot will burst when a very sudden cold snap comes through, while a strong metal well-insulated one can survive a mild winter with no issues. It doesn’t always happen at a certain temperature, and cracking the faucet may or may not work depending on variables.
The simplest experiment I know of that demonstrates this is a set of metal tubing in my yard. 3/4" pipe formed into an upward facing U-shape, open at the ends. During the summer it rains and the pipe partially fills with water. That same water froze during the winter. Despite being completely open (no-pressure) on both ends, a section of the pipe burst open when the water froze. The water froze in a way that created a plug which prevented it from expanding length-wise through the rest of the pipe.
Anyone who’s ever tried to just push a cylinder of ice out of a frozen a pipe knows how strong the friction is. Doesn’t matter if you leave a little crack in the faucet to reduce the pressure. Once ice forms across the full cross-sectional area of the pipe you’re pretty much screwed unless something warms it up before the material fails.
Dripping the water works by keeping enough of the pipe thawed in the cold exposed section via the heat of the incoming water to prevent it from freezing solid.