In general, the answer is "slowly." Augustus didn't just jump in yell "Ah ha! Monarchy!" It was the environment that let people want a single, strong leader, and it was Augustus who made sure the facade of the Republic remained so that nobody panicked.
Not to get too much into it, but, of course, the Roman fear of monarchy was alive and well. As I remember, Antony, as Consul, legislated away the Dictatorship, and it was considered a popular measure (at least for the Senate and People): https://books.google.hu/books?id=6y8RBQAAQBAJ&pg=PT72&lpg=PT72&dq=antony+legislates+away+the+dictatorship&source=bl&ots=cCWi2RkMJN&sig=gyowebWwda4EXFUcqR-Xqbn2Rgw&hl=en&sa=X&ved=0ahUKEwiLrNLVlLfQAhXGSBQKHRMNCiAQ6AEIODAF#v=onepage&q=antony%20legislates%20away%20the%20dictatorship&f=false
Right when the Second Triumvirate was set up, they were careful to hide it's "dictator-ness" (even though it totally was). It retained it's Republican veneer by it's name (Triumviri rei publicae constituendae - Triumvirate to Reconstitute the State). Also, the sharing of power between more than one person had long precedence, obviously.
Once Augustus defeated Lepidus, and later Antonius, he took on powers, always in Republican veneer, very, very slowly, over a 30 year period. First, he got himself repeatedly elected consul from 31 BC to 23 BC or so (ostensibly, to guide the Republic through difficult times). In 27 BC he pretended to step down and retire, and when the Senate clamored for him to take up his position again, he did so, but with some blandishments. He was made governor of five provinces (like Spain and Egypt) and had his control of them mandated for 10 years - this wasn't too far from the "Extraordinary commands" granted to Pompey and Caesar many years previously.
He also gained other honors and titles; Father of the Country, Imperator (victorious general), the title Augustus, and eventually was named Pontifex Maximus after Lepidus's death. After 23 he stepped down from the consulship and took a "proconsular imperium" power that spread over the whole empire. He was also given the power of Tribune for five years, meaning he could veto laws, and hurting his person was considered sacrilegious.
This went on. As you can see from this VERY short explanation, he was circumsept and slow about it. He didn't want to end up like his uncle cum father, Caesar, stabbed to death by the Senate (or a mob for that matter). So he went slowly, so slowly people probably didn't notice, and others would simply think "that's the way it should be." I read one article/quote that said Augustus himself wanted to be "like everyone's old rural countryside granddaddy."
Source: http://www.roman-emperors.org/auggie.htm (section called From Octavian to Augustus).