He didn't. To be very clear, in no way, shape or form did Augustus maintain an illusion of Republican government. There was absolutely nobody under any illusion that Augustus was effectively a monarch. If the civil wars had not made that obvious, the trimming down ("purge" for the more melodramatic) of the Senate in 28 BCE would have tipped them off. Even the thickest would have noted the execution of Murena in 22 BCE without trial, for conspiring against Augustus. Augustus ruled as a monarch, behaved as a monarch and was treated as a monarch.

Now, that being said, the "facade of the Republic" simplification comes from the way that Augustus was very careful not to interfere with or diminish the status and duties of the Senate. Senators remained extraordinarily influential people, they commanded many of the most prestigious provinces, they had their traditional honors and Augustus called on them very often for consultation. The average Senator, in fact, probably would not have been terribly affected by the change, after all, in practice very few of the members stood a chance at imperial office or a major voice on policy during the Republic anyways.

Outside of the Senate there was even less ambiguity. The neighborhood organizations of the city of Rome (vici) took Augustus as their patron, and foreign leaders had no trouble distinguishing who was really in power. Imagery of Augustus was extremely imperial, to the point where the famous Temple of Augustus in Pula/Pola was certainly started during his lifetime. In this contexts, Senators effectively became the "aristocracy" of the monarch, but there was no more confusion as to Augustus' status tan to Louis XIV's.