Julius Caesar is a tragedy, which means that the play follows the downfall of a tragic hero. A tragic hero is a character that experiences failure as a result of a flaw in his/her personality. Many critics disagree about who the tragic hero of Julius Caesar is (could be Caesar, could be Brutus), but many of the characters are alike in their arrogance. The significance of the omens are to highlight the arrogance.
Let's start with Caesar. Not only is Caesar approached by a soothsayer and told to beware the Ides of March, his own wife has a troubling dream that he will die on this day. Caesar believes enough in the possibility of these things that he asks for a ritual sacrifice to be done to bring further news - and learns that he is unlucky on this day. He at first agrees to stay home, but a little flattery and a suggestion that he is cowardly makes him quickly change his mind:
Decius Brutus: “Break up the Senate till another time,
When Caesar's wife shall meet with better dreams.”
If Caesar hide himself, shall they not whisper
“Lo, Caesar is afraid”?
If Caesar had not been so proud and so arrogant - had he not been so concerned with how he might appear to others - he would not have been in front of the conspirators.
Next, lets look at Brutus. After already making the tragic mistake of killing Caesar, Brutus is faced with decisions that could alter his future fate. One such decision is presented by Cassius, who suggests that they not wear out their soldiers by marching forward to meet Antony and Octavius in battle. Brutus disagrees and insists they go forth to Phillipi. After, Caesar's ghost appears to Brutus and says he will see Brutus at Phillipi - suggesting that Brutus will die there. Brutus recklessly ignores this warning, and goes forth anyway. Like Caesar, he dies where the warning suggested he would.
Even Cassius sees the omen of the ravens to warn him before entering into battle. Cassius admits at this time to being scared of the omen, but he does not change his own course to protect himself.
The omens, we see, show us that these men are either too arrogant or too stubborn to consider changing their course at the suggestion of others.