The purpose of the interchange between Murellus and the commoners in the opening scene of Julius Caesar is primarily to establish a sense of civil unrest revolving around Julius Caesar. He is loved by the commoners but hated and feared by many of the upper-class Romans because they consider him a power-hungry demagogue who aims to become absolute dictator or even king. Shakespeare has chosen to present the confrontation as partly comic by having the Cobbler make Murellus look foolish. The Cobbler equivocates with the tribune in the same way that the Sexton does with the Prince in Act 5, Scene 1 of Hamlet.
Besides establishing the disorderly and potentially volatile spirit of the times, the opening scene of Julius Caesar quickly captures the attention of the audience and utilizes some of the extras Shakespeare needed to employ for his big, pivotal scene in which Mark Antony's eloquence turns the lower-class Romans against the men who assassinated Caesar.
Shakespeare must have had at least two dozen extras on the premises who were being paid to do nothing more than shout and gesticulate. It is noteworthy that the Cobbler does most of the talking for the Commoners. Shakespeare apparently decided to write two additional scenes for the play in which he could employ these supernumeraries. One is the opening scene in which the tribunes confront the commoners, and the other is the scene in which the riotous mob tears the hapless Cinna the Poet to pieces just because he happens to have the same name as one of the conspirators (III.3).
Although Murellus and Flavius are successful in driving the commoners off the street, we learn later that both of them have been "put to silence." According to a footnote in my edition of the play, this means that they were deprived of their tribuneships and exiled--but they might have been murdered on Caesar's orders. They never appear again in the play and are never mentioned after Act I.2.