In Act I, Scene 2, it is the Feast of Lupercal and Brutus and Cassius watch Caesar in a procession through the streets of Rome as the crowd shouts for him. Brutus expresses concern that the Romans may wish to make Caesar their king; Cassius responds that he believes that if Brutus is anxious about Caesar attaining such power, he would "not have it so," and he launches his persuasive arguments upon the premise that it is, indeed, honor about which he wishes to talk with Brutus.
In lines 138-188, then, Cassius fuels the anxiety of Brutus that Caesar may become tyrannical if given too much power, likening Caesar to the Colossus which towers over men. Further, he tells Brutus that men sometimes miss opportunities to change things because they do not act:
The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings, (1.2.146-147)
He suggests that the name of Brutus can follow the sound of trumpets just as easily as the name of Caesar:
O, you and I have heard our fathers say,
There was a Brutus once that would have brooked
As easily as a king. (1.2.186-188)
The allusion here is to Lucius Junius Brutus, who helped to expel the last king of Rome and worked to found the Republic in 509 B.C.
The persuasiveness of Cassius derives from his ability to take the anxieties of Brutus and use them as the foundation for his argument; namely, his fear that the Romans will make Caesar king, an act which will afford Caesar so much power that he will become corrupt and tyrannical.