Why does Shakespeare include the attack on Cinna the poet in Act 3 of Julius Caesar?

There are two reasons Shakespeare includes the scene.  The first is to show the mentality of the mob following Antony's speech.  This group of people is so intent on burning down houses and exacting revenge that they are beating and killing people just for having the same name as conspirators.

The second reason is the comedic relief that they scene adds.  I know it seems odd to talk about comedy in a scene where an innocent bystander is torn apart, but "tear him for his bad verses" is pretty funny.  Shakespeare does this in many of is tragedies, includes a funny scene either just before or just after a horrible act.  In this case, it occurs after the killing of Caesar.  This was Shakespeare's way of easing the tension of the audience.  WS didn't think an audience would enjoy an entire production full of suspense and horror, so he gave them short scenes where they could giggle and relax a little bit.

What does the attack on Cinna the poet in "Julius Caesar" signify?

The unfortunate Cinna becomes a victim just because he has the same name as one of the conspirators. He is desperate to prove that he is not one of them, but his pleas fall on deaf ears. The crowd is driven not by reason, but by rage.

Two aspects are important in this instance. We know, in the first place, that there is a conspirator called Cinna. He converses with Cassius in Act 1, Scene 3, and they are discussing the portentous weather conditions and Cassius's attempt to win Brutus over to their nefarious cause. Cinna says, in part:

CINNA
Yes, you are.
O Cassius, if you could
But win the noble Brutus to our party—

He is clearly involved in the plot and follows Cassius's instructions to place papers urging Brutus to join the conspiracy in the praetor's chair, on Brutus' statue, and in his chambers where he will surely find at least one of them.

Furthermore, the crowd has already been uneasy after learning about Caesar's death and are incensed and driven to insurrection by Antony's powerful speech at Caesar's funeral. Before Antony's address to the crowd, Brutus tells them about the necessity of having Caesar removed. He appeals to their reason, and they easily accept his explanation that Caesar's assassination is for the good of Rome.

Antony's brilliant speech, however, appeals to the upset throng's emotions. He gradually works the citizens into a frenzy by carefully delaying their desire to rush out and seek vengeance. He deliberately uses sarcasm, irony, innuendo, and rhetorical questions when he refers to Caesar's goodness and contrasts it to the reprehensible abomination that has been committed. The mass starts baying for blood, and he only sets them free once he knows that they are unstoppable and will go out to wreak havoc. At the end of his speech, Antony says:

Now let it work. Mischief, thou art afoot,
Take thou what course thou wilt!

The crowd is so blinded by anger and a lust for retribution that everything and anyone that seems suspect becomes fair game. It is in this situation that the poor Cinna the poet finds himself when he is confronted. The irrational mob criticizes him for his name and his bad poetry and decides to tear him to pieces.

Fourth Citizen
It is no matter, his name's Cinna; pluck but his
name out of his heart, and turn him going.

Third Citizen
Tear him, tear him! Come, brands ho! fire-brands:
to Brutus', to Cassius'; burn all: some to Decius'
house, and some to Casca's; some to Ligarius': away, go!

This happens in Act 3, Scene 3, when the people of Rome are really quite angry because of the assassination of Caesar.

The fact that the people are angry can be shown by their first reason given for wanting to kill Cinnna. Cinna was the name of one of the conspirators. The fact that they want to kill him shows they are angry about Caesar's death.

The fact that they kill him anyway even after they find out he is not the "right" Cinna shows they are out of control.

Why does Shakespeare include Cinna the poet's death in Julius Caesar?

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. 

Shakespeare was not only a playwright but a director, producer, actor, and co-owner of the theater. His production of Julius Caesar required a mob of extras to play the plebians who are turned into a rampaging mob by Mark Antony's great funeral oration. No doubt, Shakespeare decided to get additional use out of these extras rather than having them sitting around during the performance doing nothing and getting in everybody's way. He uses some of these extras in the very first scene of the play, when they are playing working men taking a holiday and are confronted by the two tribunes, Flavius and Marullus. It can be seen that only one of this group of men has a speaking part in Act I, Scene 1. That is the actor who plays the Cobbler, who is undoubtedly a professional actor and a member of Shakespeare's company. Shakespeare gets additional use from the extras by inserting a scene in which the rampaging mob confronts Cinna the poet and tears him to pieces because he has the same name as one of the conspirators. The anecdote about Cinna is contained in Plutarch's Life of Julius Caesar. It includes this sentence:

For there was among the conspirators man who bore this same name of Cinna, and assuming this man was he, the crowd rushed upon him and tore him in pieces among them.

Shakespeare utilizes the incident to demonstrate the fury of the Roman mob and to create a strong impression on his audience. According to Plutarch (but not to Shakespeare) it was this incident involving the innocent Cinna that made Brutus and Cassius decide to flee the city. 

No doubt some of the extras who appeared in the three scenes mentioned above would also represent soldiers on both sides at the Battle of Philippi in Act V.