Shakespeare uses storms to portend or foreshadow coming events and to set the tone or atmosphere for a scene. As other Educators have noted and discussed, the storm in act 1, scene 3 of Julius Caesar symbolizes and portends the coming storm in Rome, foreshadows the disturbance and chaos that will be caused by Caesar's assassination, and sets the tone of secrecy and uncertainty for the conspiracy scenes leading up to the assassination.
The sudden flashes of lightning and the loud thunder at the beginning of the scene seem a bit heavy-handed, unless Shakespeare intended to wake up anybody in the audience who dozed off during the previous scene and alert everybody else in the theatre to pay attention to the growing conspiracy against Caesar that follows in the next two scenes.
Judging by Casca's lines in the early part of the scene, it's apparent that Shakespeare wanted to emphasize the conspiracy and the coming disruption in Rome, and he wants to make it perfectly clear that's exactly what the storm means.
When Casca meets Cicero at the beginning of the scene, Casca is shaking in his sandals.
CICERO. Why are you breathless, and why stare you so? (act 1, scene 3, line 2)
Casca has never seen a storm like this before.
CASCA. Never till tonight, never till now,
Did I go through a tempest dropping fire.
Either there is a civil strife in heaven,
Or else the world too saucy with the gods
Incenses them to send destruction.
(act 1, scene 3, lines 9–13)
The storm portends "civil strife" and "destruction."
On the other hand, Cicero seems to be enjoying the storm.
CICERO. Why, saw you anything more wonderful? (act 1, scene 3, line 14)
Cicero unknowingly voices the intentions of the assassins to make Rome a better place.
At this point, however, Casca leaves the real world far behind as he tells Cicero what he's seen this evening. It appears that Casca has been eating from the insane root that Banquo mentions in Macbeth, "That takes the reason prisoner" (Macbeth, act 1, scene 3, lines 87–88), because Casca has clearly been hallucinating—or he was struck by lightning.
CASCA. A common slave ...
Held up his left hand, which did flame and burn
Like twenty torches join'd ...
Against the Capitol I met a lion,
Who glaz'd upon me and went surly by
Without annoying me: and there were drawn
Upon a heap a hundred ghastly women
Transformed with their fear, who swore they saw
Men all in fire walk up and down the streets. (act 1, scene 3, lines 15–28)
Casca finally gets to the point that Shakespeare is trying to make with the storm and Casca's crazy-talk.
CASCA. When these prodigies
Do so conjointly meet, let not men say
“These are their reasons; they are natural,”
For, I believe, they are portentous things
Unto the climate that they point upon. (act 1, scene 3, lines 29–33)
There it is! "Portentous things ... that they point upon." Shakespeare couldn't be any clearer than that.
Cicero stands looking at Casca in stunned amazement for a moment, then he gathers his thoughts and puts Casca's ravings into perspective.
CICERO. Indeed, it is a strange-disposed time;
But men may construe things after their fashion,
Clean from the purpose of the things themselves. (act 1, scene 3, lines 34–36)
Cicero can't wait to get away from crazy Casca, and he changes the subject.
CICERO. Come Caesar to the Capitol tomorrow? (act 1, scene 3, line 37)
With his portentous foreshadowing accomplished, and with some semblance of reality restored to the scene, Shakespeare sends Cicero away but not without giving him a great exit line:
CICERO. This disturbed sky
Is not to walk in.
(act 1, scene 3, lines 40–41)
The conspirators gather and get down to business under a "disturbed sky," with strategically-placed lightning and thunder.