When Octavius/Augustus became the first "emperor" of Rome, Rome had been a republic for five hundred years, with elected leaders and no tradition of public offices being inherited dynastically. In fact, they had a strong mythic memory of a overthrowing a tyrannical king, and great pride in having no kings thereafter. - Just like the USA does now. I'd expect the Romans to reject the idea of a hereditary leadership position. Inheritance law only applied to private property. It should have sounded like the imperial family was treating the whole empire like they owned it, like one big estate.
And then, I could imagine maybe Augustus was so canny that he pushed his own heir through, but his dynasty was so horrible that I'd think they should have totally discredited the whole idea. But they didn't. Some hundred years later, this idea of inheriting was so strongly established that emperors would go through the legal form of "adopting" an heir to satisfy the idea of inheriting the emperorship. (Imagine one US president having to adopt the next one.)
How did this happen? How did they accept it? How did they understand it?
There are a couple reasons why the Romans accepted the new system. Firstly, bloodlines were always important to the Romans, even during the Republic. A Roman’s auctoritas included his ancestors’ accomplishments as well as his own. For example, Brutus was well aware that his ancestor played a pivotal role in the banishment of Rome’s last king, Tarquinius Superbus and felt compelled to live up to that reputation. Adopted sons, such as Scipio Africanus Aemilianus, had the auctoritas of both their biological and adoptive families. Though they liked to pretend it was a merit-based system, the Romans had always believed in familial importance.
Also, we need to consider the circumstances of Augustus’s rise to power. During the final century of the Republic, Rome endured the civil wars of Sulla and Marius, the proscriptions of Sulla, the civil wars of Julius Caesar and Pompey, the civil war of the second Triumvirate and the Liberators, the proscriptions of the second triumvirate, the war between Sextus Pompey and Octavian, and finally the civil war between Antony and Octavian. The backdrop of these wars was the complete destabilization of the Roman political system. Violence and bribery were so rampant that they became the norm, as we saw with the Gracchi brothers. Rome had no major enemies and more wealth than ever before, leading to cyclical internal strife. Throughout this period, the patrician class was also significantly reduced and loyalty to the Republican system was proportionally minimized. So when Octavian won the final civil war and introduced 30 years of peace and stability, the Roman people were largely grateful. Many of the benefactors of the old Republican system had been killed off or replaced in the senate by men loyal to Octavian. Few of the surviving Romans even remembered what a functioning Republic looked liked. All they remembered was the century of civil wars.
Augustus was also a master politician, who was careful not to show his dictatorial powers. Whereas Julius Caesar had himself named dictator for life, Augustus simply adopted a variety of political powers that would give him the same authority. He wasn’t a king, merely a princeps.
Lastly, the Romans were well aware of potential power vacuums during transitions of power. When Julius Caesar was murdered, the Liberators seemed to believe that Rome would simply return to the Republic it had been. Instead, Caesar’s key lieutenants and adversaries fought to reclaim power. No one wanted to return to more civil wars.
As Augustus’s power was entirely unique, no one was sure how a transition back to a true Republic would work. The new princeps system clearly worked better, so why even go back? Instead, he handed the reigns over to Tiberius who had the pedigree and auctoritas to rule competently. At least for the first twelve years of his reign, that’s exactly what he did.