Why does the crowd kill Cinna the Poet in Act III of Julius Caesar? What is Shakespeare saying about the Roman public?

The crowd mistakes Cinna the poet for Cinna the conspirator, which shows the Romans are whipped into a frenzy and not really paying attention to details at that point.
There was a conspirator named Cinna. Unfortunately, poor Cinna the poet is mistaken for this other Cinna after Mark Antony turns the Roman citizens into an angry mob with his funeral speech. He stirs up anger and resentment toward the conspirators. By calling Brutus and the conspirators honorable men in one breath and calling them murderers in another, Antony tells the crowd he does not want mutiny while basically telling them to mutiny.

Show you sweet Caesar's wounds, poor poor dumb mouths,And bid them speak for me: but were I Brutus,And Brutus Antony, there were an AntonyWould ruffle up your spirits and put a tongueIn every wound of Caesar that should moveThe stones of Rome to rise and mutiny (Act III, Scene 2).

After the speech, a group accosts Cinna the poet on the street and begins to interrogate him. He is confused because he was just innocently walking along. They ask him his name and where he is going. Although he tells them he is not a conspirator, they decide to kill him anyway.

CINNA THE POET
Truly, my name is Cinna.
First Citizen
Tear him to pieces; he's a conspirator.
CINNA THE POET
I am Cinna the poet, I am Cinna the poet.
Fourth Citizen
Tear him for his bad verses, tear him for his bad verses (Act III, Scene 3).

They want to kill Cinna when they think he is a conspirator. When they find out he is not, they want to kill him anyway; they are in such a frenzy that they just want an excuse to kill anyone. There is no reason to kill a poet for bad poetry. This crowd probably hasn’t even read his poetry. They are just out for blood.
Shakespeare's point about the people of Ancient Rome is that they are so stirred up by this point that they are bloodthirsty. This is why they kill an innocent man. They are a weapon and Antony loaded and pointed them. He understands that the people of Ancient Rome are a little wild sometimes.