At the most obvious level, the civil war establishes the context in that the whole play emerges out of war (therefore, this hints that there will be more war to come). At a more interesting level, the opening civil war nicely dovetails with the civil war that Macbeth's murder of Duncan ultimately sparks.
The parallels between the two are interesting: Duncan is a good king who inspires trust, yet he is twice the victim of traitors (first, the Thane of Cawdor, then Macbeth, the new Thane of Cardor). When Macbeth falls victim, it is not due to a traitor, but is a direct result of his own actions.
The war also sets up the differences in the two Kings' styles--Duncan goes to war to maintain order; Macbeth, to disrupt it. Were it not for the forces of darkness, Duncan would have ruled his country with an even hand. Not so, Macbeth. He rules as a tyrant because he began unlawfully. Even though he is the king, he is like the opposing army because his rule batters at Scotland's peace.
By starting the play with the civil war, Shakespeare alerts the audience that what follows will be a battle between two opposing sides, the rightful heirs (good) and the pretenders (evil). The audience knows to expect strife, bloodshed and bravery. We also learn that there may be some problems with trust, with determing who or what is trustworthy. This theme is established early, through Duncan's innocent trust of both Macbeth and Cawdor, and continues later, in Macbeth.