Shakespeare drew heavily on Plutarch for his play Julius Caesar. The supernatural elements in the play are all events reported by Plutarch, mostly as rumor and hearsay. Here are some good examples from Calpurnia's dialogue in Act 2, Scene 2:
There is one within,
Besides the things that we have heard and seen,
Recounts most horrid sights seen by the watch.
A lioness hath whelped in the streets;
And graves have yawn'd, and yielded up their dead;
Fierce fiery warriors fight upon the clouds,
In ranks and squadrons and right form of war,
Which drizzled blood upon the Capitol;
The noise of battle hurtled in the air,
Horses did neigh and dying men did groan,
And ghosts did shriek and squeal about the streets.
The main appeal of Shakespeare's Julius Caesar is that it takes the audience back in time, making them time-travelers. Through Shakespeare's hypnotic word-magic the audience is back in Rome in the year 44 B.C. The audience may not believe in such portents as Calpurnia describes, but they make the various characters seem real, since most Romans did believe in the reality of such supernatural phenomena. They truly believed in the Roman gods, who had been derived from the earlier Greek gods. A modern audience would be even less likely than an Elizabethan audience to believe in such supernatural phenomena, but a modern audience can be transported back in time just like an Elizabethan one. We find ourselves giving credence to these fierce fiery warriors fighting upon the clouds and ghosts shrieking and squealing about the streets because we are back in Rome in 44 B.C. After all, when in Rome do as the Romans do.
The supernatural phenomena are also very effective in foreshadowing the assassination of Julius Caesar. All these strange occurrences must mean something. As Calpurnia tells her husband in beautiful Shakespearean language:
When beggars die, there are no comets seen;
The heavens themselves blaze forth the death of princes.
Shakespeare uses two acts to build up tensions and expectations. The audience expects the high point of the play to be reached when Julius Caesar is murdered in the Capitol. All the supernatural events contribute to this tension and these expectations. However, the actual assassination is a letdown, a disappointment. It is anticlimactic. Here is how the text describes it.
They stab Caesar.
The glory of Shakespeare's plays is never in the action but always in his poetic language. The audience may feel disappointed with the emotional effect of Caesar's assassination, but they will soon be fully compensated when Antony tentatively and rather awkwardly begins his funeral address with
Friends, Romans, countrymen, lend me your ears.
Antony's funeral oration is probably the greatest thing Shakespeare ever wrote. Not only that, but it is a turning point in the play, as well as a turning point in history. Antony is facing a hostile audience, but he turns them completely around and starts a mutiny that forces Brutus and Cassius to flee from Rome. All the supernatural occurrences in the first two acts were not intended to serve as a buildup to Caesar's assassination but as a buildup to Antony's funeral oration and its aftermath. Rather than foreshadowing Caesar's death, they might be said to foreshadow the birth of the Roman empire inspired by the indomitable spirit of the great Julius Caesar.