In addition to highlighting the mob mentality, the cry "Kill him for his bad verses" is also Shakespeare's way of interjecting a little humor into a very tragic act. This is something he does in many of his tragedies to lighten the mood either before or after one of the most tragic scenes in the play.
The event was recorded by Plutarch, and Shakespeare was following Plutarch closely in writing his play. As a practical matter, Shakespeare had to hire a lot of extras for this play because he needed men to play the mob listening to Brutus and then listening to Antony at their funeral addresses for Caesar. Since Shakespeare had all these extras on hand, as well as underfoot, he apparently wanted to make more use of them. He opens the play with a group of working men taking the day off to celebrate Caesar's triumphs. These are undoubtedly the same extras as those who played the mob members at Caesar's funeral. Then Shakespeare must have decided to make further use of them in the short scene in which they kill the unfortunate Cinna the poet. It can be observed that most of the men do not have speaking parts, probably because they are just loiterers picked up off the streets for single performances. No doubt the mob that is heard offstage roaring their enthusiastic support of Julius Caesar while Brutus and Cassius are talking onstage are the same extras, and some of them were probably given armor and used to play soldiers at the battle of Philippi.
Another reason Shakespeare had for inserting that scene involving the death of Cinna was to show that there was widespread rioting in Rome and that this was how things became chaotic when the masses, the "Great Beast," as Thomas Jefferson called them, got out of control. The same thing happened during the French Revolution. Shakespeare knew that the ignorant, exploited masses were always a danger to civilization. Once they began looting and killing, there was no way to stop them. Brutus was afraid there would be chaos, and Antony predicts there will be chaos following the death of Caesar.
Over thy wounds now do I prophesy
Which like dumb mouths do ope their ruby lips
To beg the voice and utterance of my tongue,
A curse shall light upon the limbs of men;
Domestic fury and fierce civil strife
Shall cumber all the parts of Italy;
Blood and destruction shall be so in use,
And dreadful objects so familiar,
That mothers shall but smile when they behold
Their infants quarter'd with the hands of war;
All pity choked with custom of fell deeds,
And Caesar's spirit ranging for revenge,
With Ate by his side come hot from hell,
Shall in these confines with a monarch's voice
Cry “Havoc!” and let slip the dogs of war,
That this foul deed shall smell above the earth
With carrion men, groaning for burial.
And when Antony finishes his funeral oration, sending the mob off to seek the conspirators and burn their houses, he says to himself:
Now let it work. Mischief, thou art afoot,
Take thou what course thou wilt.
Brutus wanted peace and order. Antony wanted rioting and bloodshed.
He was trying to forewarned a tragic event that was going to happen in the story and it was portrayed accurately as later on, Julius Caesar was killed knowingly by a group of conspirators. It was trying to add a bit of a comic relief towards a doom and gloom situation and lighten the tense atmosphere