Essentially by placing himself at the head of a extraordinarily stable power structure, which left very few incentives for a revolt. While part of that was promises of money and land, there was also a command structure down through the armies which Augustus carefully designed - especially via his own proconsular powers in the provinces. The armies weren't scattered, but carefully placed under his commanders and with all their victories accredited to him. At the same time he very successfully pacified much of the Empire, so there wasn't a situation like the later crises, with various regional discontented armies holding the only effective power - he essentially managed to become both the central and provincial authority.

Augustus was always very careful not to be caught in a situation where he might have to disappoint the soldiers - lessons from the immediate aftermath of the founding of the Second Triumvirate where he failed to peacefully settle the veteran soldiers. See the Res Gestae 15, for a record of his repeated gifts. His power and riches far outstripped everyone else and therefore there was little threat of the soldiers supporting possible rivals.

However more important is probably Tacitus' point that he 'seduced everyone by the delights of peace'. He was essentially seen as having secured the frontiers of the Roman world. He had very capable generals under his command, most notably Agrippa. With the settlement of 27 BC he gained proconsular power over Gaul, Spain, Syria and Egypt and their accompanying armies. He also was granted legal power over all the other proconsuls as well which meant he was the only one who could receive triumphs and he was legally the head of every Roman army. Many of the commanders on the frontiers were of his own selection - obviously certain members of his own family such as Tiberius. The achievements of individual generals were downplayed, avoiding the competition for glory that was a feature of the previous generation (for example Marcus Linius Crassus the Younger was denied war trophies for defeating the leader of the Bastarnae tribe in single combat, though he was given a triumph).

He also had his own personal Praetorian Guard of 9,000 in Rome and Italy (3,000 in Rome, the rest garrisoned throughout Italy), who at this point were personally loyal to Augustus and not the source of the kind of unrest you saw in earlier times.

Essentially Augustus' time was very different from the earlier and later reasons for unrest in the army. Earlier, the race for glory led to personal military expeditions - something Augustus´superior proconsular power dampened. Later it was the loss of centralised control that led to regional armies claiming power, especially the Praetorian Guard. This wasn't the case under Augustus who tightly held centralised control and often visited and led campaigns in the regions.