Does running water out of a faucet prevent the pipes from bursting? If so, why? I have been told that, during especially cold periods during winter, one should run water out of the tap to ensure the pipes do not burst. 
Does this really help? If so, why? If true, at what temperature should one start to adopt this policy? 
 A: Yes this does really help. 
Due to the density anomaly of water it expands even when it's frozen and in a solid state (ice). This causes your pipes to burst over the time due to the increased pressure. Letting the water dripping a little bit serves as a pressure relief and therefore prevents the pipes from bursting. It is important to run both, hot and cold water through the faucet if you have the option to run them seperatly. 
You should adopt that policy once the temperatures drop well below the freezing point of water. At which exact temperature I can't tell you. It depends on a lot of factors: how well are you pipes isolated, etc.. For a garden hose for example you should drain all the water before the temperaturs drop to the freezing temperature of water. You might want to contact a local plumber service, maybe they have experience at what temperatures they have been called to fix broken pipes.
Further reading:
http://www1.lsbu.ac.uk/water/water_anomalies.html
http://www.redcross.org/prepare/disaster/winter-storm/preventing-thawing-frozen-pipes
http://www.nola.com/homegarden/index.ssf/2014/01/you_know_the_drill_run_the_pip.html
A: Moving water doesn't freeze as fast as still water.  Boaters who leave their boats docked in bodies of water that freeze over install bubblers at the dock to keep the water moving. The effect is that the immediate area around their boat doesn't freeze over.   It's the same principle for running water through you faucet to prevent a freeze.
That is part one of the answer.
Part 2 in the case of running water through a faucet is that your supply pipe is always buried below the frost line.   In other words water in the supply pipe is above freezing at all times.  Keeping the faucet running brings this higher temperature water through your pipes helping to prevent a freeze.
