I would say that its dramatic significance or purpose is three-fold:
- To set the play in time and place and introduce the character of Caesar through the opinions that common citizens have of him versus the opinions held by two Roman tribunes. The purpose here is exposition.
- It also serves to immediately involve the audience as "actors" in the performance. Julius Caesar, almost more than any other play by Shakespeare, relies upon characters directly addressing the audience, including them in the action as citizens -- whether peaceful ones or an angry mob -- of Rome. This was a common stage practice in Shakespeare's theatrical world. Audience involvement by the actors onstage was an expected Dramatic Convention and is set up here, at the very opening of the play as Flavius and Marullus direct their lines to the audience as well as the actors playing the Commoners.
- The opening scene also provides an inviting comic doorway into the play, which is one of Shakespeare most stringent, in that there isn't much comic relief from the dramatic action. Here, in the opening, he offers his acting company's clowns (comic actors) the chance to "warm the audience up," playing the Commoners in this scene to the straight men Flavius and Marullus.
In terms of the plot of the play, the opening scene also serves to show how differently the Roman populace saw Caesar from some of the more noble Roman citizens, and how easily the ordinary commoner's mind can be changed. Enotes has a great analysis of this scene in its Julius Caesar Study Guide, which I've linked below.