The Mandate of Heaven was essentially a legitimization tool. By stating that their leadership was not just because they were selected or part of some royal family but instead because of divine choosing greatly strengthened their authority. For example, imagine the emperors before the Mandate of Heaven was established. Besides the obvious fact that they were emperor, there's no real reason that the people should follow their rules (at least no significantly convincing one). Now, on the other hand, the people believed that those rules and demands by their emperor were actually divine.