How did the Tang people gain the Mandate of Heaven?

The tang invade from the north and overthrew the Sui dynasty therefore gaining the mandate of Heaven.

The Mandate of Heaven has a similar concept of a checklist. If a dynasty does not fulfill the obligations needed, then they lose the Mandate of Heaven; the right to rule. Signs that someone lost the Mandate of Heaven are droughts, invasions from foreign countries, earthquakes: generally any sort of disaster. For the Tang dynasty, the An Lushan rebellion pretty much destroyed them and their prosperity. It left the dynasty weakened, and from there they never regained their former glory.