The short answer to this question is that nothing happened. The process of Rome becoming an empire is widely misunderstood and it took a long time. To begin with, one might easily argue that, in terms of territory, Rome had become an empire long before Octavius Caesar was even born. It was an empire without a emperor, saddled with a form of Republican government that had worked well enough for a city state, or even a country, but was no longer fit for purpose.
Octavius Caesar himself was far too astute a politician ever to claim a title which suggested royalty, as the word "emperor" does to us. He had many titles. The name by which he has generally come to be known, Augustus, was itself a title, but the closest to "emperor" was "Imperator," a military role which we might translate as "Supreme Commander."
The upshot of this is that the average person who lived in Augustan Rome would have been quite unaware of any definite transition from Republic to empire. They would probably have said that the Republic fell into a brief period of turmoil after the assassination of Julius Caesar, and that his son (for the Romans took adoption very seriously) had restored it to its former glory.
Of course, Augustus's gradual reorganization of the state meant that life did become more peaceful and orderly under the Pax Romana, and that this was largely true in the wider empire even after his death, when the city of Rome itself suffered from the tyranny of Caligula or Nero. However, this was a slow process, unfolding over decades of Augustus's rule.