The group of Roman lower-class citizens is ordered to disperse by the tribunes Flavius and Murellus because they are all out honoring Julius Caesar. The tribunes are members of the upper class and are opposed to honoring Caesar because he is becoming too powerful by favoring the working people at the expense of the aristocracy. Some of the dialogue is especially pertinent.
But indeed, sir, we make holiday to see Caesar, and to rejoice in his triumph.
Wherefore rejoice? What conquest brings he home?
What tributaries follow him to Rome
To grace in captive bonds his chariot wheels?
Shakespeare opens the play by dramatizing the animosity that exists between the rich and the poor in Rome in these turbulent times. This is a good way of securing the attention of Shakespeare's audience and obtaining silence in the auditorium so that everyone can hear the actors.
Shakespeare must have had another reason for opening with a mob scene. He was going to need a large number of extras for the big scene in Act III in which both Brutus and Antony deliver their funeral speeches and Antony starts a riot. Since Shakespeare had recruited a number of men--probably straight off the street--and had to pay them for the full day, he could make additional use of them in this opening scene. Most of them have no lines. The Cobbler is a professional actor who does most of the talking for the whole group.
So Shakespeare uses these extras again in the spectacular Act III, Scene 2, and then he makes a little further use of them in Act III, Scene 3, in which the mob kills Cinna the poet just because has the same name as one of the conspirators.
Tear him for his bad verses, tear him for his bad verses.
I am not Cinna the conspirator!
It's no matter, his name's Cinna.
Flavius and Murellus succeed in dispersing the citizens in the opening scene of the play, but in Act I, Scene 2, Casca tells Brutus and Cassius:
Murellus and Flavius, for pulling scarves off Caesar's images, are put to silence.
It seems most likely that Casca means Caesar has secretly had the two tribunes murdered.