Octavius Caesar is concerned about arriving in Rome after Caesar’s death because he thinks the conspirators will kill him next.
Octavius was Antony’s grand-nephew, and was officially named his son by adoption in his will. This is when he added the “Caesar” to his name. It was actually quite common in those days to posthumously adopt a son and heir. However, by adopting Octavius, Caesar also made him a target.
At the time of Caesar’s death, Octavius was on a training exercise and staging for a military campaign. He learned of Caesar’s death on his way to Rome. Almost immediately after Caesar’s death, Antony began the machinations to take control. One of the first things he did was send a messenger to Octavius telling him that it was not safe for him in Rome.
In some ways, Antony was not wrong. Octavius Caesar was a potential target. As the son of Julius Caesar and the heir to his name, his fortune, and his will, Octavius also had a chance to gain all of Julius Caesar’s political supporters. If the conspirators were going to kill Caesar's family, he would definitely have been next. He suddenly became the richest young man in Rome.
However, this is also a reason for Antony to keep young Octavius Caesar out of Rome. Even though he was young, he was the heir of Julius Caesar and had his name. If he was in the city and the one delivering speeches, he would also be the one getting the support of the people. For this reason, Antony asked him to remain outside the city, and that gave him time to give his famous speech, in which he could produce Julius Caesar’s will and get the people to throw their support to him.
But here's a parchment with the seal of Caesar;
I found it in his closet, 'tis his will:
Let but the commons hear this testament--
Which, pardon me, I do not mean to read--
And they would go and kiss dead Caesar's wounds
And dip their napkins in his sacred blood,
Yea, beg a hair of him for memory,
And, dying, mention it within their wills,
Bequeathing it as a rich legacy
Unto their issue. (Act 3, Scene 2)
By telling the people that Caesar left them money, Antony is making himself Caesar’s political heir in their minds. He is hoping that he will get the people’s support. He does. They throw their support behind him almost instantly, instead of the conspirators, before Octavius Caesar is even in the picture.
Antony needs Octavius Caesar though. He cannot maintain the people’s support, and the army’s, on his own. Octavius has Caesar’s name, his money, and a great deal of the support of his followers. Antony and Octavius Caesar thus form a triumvirate with Lepidus, a sort of third wheel from Caesar’s army command.
Although Octavius is young and lacks Antony’s military experience, he does not just bow down to the older man’s will. The two of them argue and jockey for supremacy almost constantly.
Octavius, I have seen more days than you:
And though we lay these honours on this man,
To ease ourselves of divers slanderous loads,
He shall but bear them as the ass bears gold,
To groan and sweat under the business,
Either led or driven, as we point the way;…
Even on the battlefield, the two continue to argue, even though they are leading two different armies working together. Octavius Caesar is brilliant and cunning, even though he is young. He sees Antony’s weaknesses and catalogues them in his mind, aware that one day he will get the better of the older man. It takes some time, and another civil war, but he does. It makes him the leader of Rome, and its first emperor.
Octavius Caesar’s situation during the early days after Caesar is assassinated is indeed a precarious one. However, as we know, the conspirators were not planning to kill anyone other than Caesar. There was no way for Antony, Octavius Caesar, or anyone else to know that though. It makes perfect sense for Antony to assume they would kill Octavius Caesar, just as he assumed they would kill him until they swore they wouldn’t. They said nothing about Caesar’s heir at that meeting. Antony made the best of a bad situation, hoping to turn it to his advantage.