Numerous factors came into play but basically it was a combination of wealth, legacy and approval. With regards to wealth, after being adopted by Julius Caesar he acquired Caesar's extensive wealth which essentially allowed him to finance many projects within the city of Rome. This was important because it meant the citizens of Rome viewed him as a sort of patron thus basically became indebted to him. The second component was legacy and by legacy I means regarding Caesar. Being Caesar's adopted son made Augustus very popular with the legions and the plebes of Rome. After the defeat of Anthony, the legions were basically under Augustus' sole control and though he didn't really go through the effort of threatening the senate but they probably already knew anyway. The love of the plebes also meant that they basically insisted on Augustus' participation in state politics. Various riots broke out between 21, 20 and 19 BC after the plebes thought that Augustus had been excluded from the consular elections. Augustus' manipulation of state politics seems to have evaded most of them. And that comes to the last point of approval, Augustus ruled through the approval of the people of Rome and the senate or atleast that's how he made it seem. In truth Augustus had stacked much of the senate with his own supporters meaning it was easier to amass ultimate power. Now the issue here was most of the aristocracy were likely aware of Augustus' Supreme power but they did little to stop him, some did try but failed, most didn't because at that point they had been tired of all the Civil wars that plagued the republic prior to that and many were afraid of power falling into the hands of multiple men again who would then likely cause more civil wars( the year of four Emperors seems a true testament to this) so they deiced it was probably best to just leave Augustus as sole ruler.

Eventually when Augustus died, most Romans had been under the principate for so long that the desire to go back to the republic was remarkably low. So in short Augustus became first Emperor through clever manipulation of politics and the public.