It seems that there was a lot of things that happened before the beginning of this play. Why would the play start here and not with the battle that killed the brothers instead.
The first part of this answer comes from understanding that Greek theatre festivals did not feature original stories until much later. Instead, playwrights chose stories that were well known by their audiences so as to explore perspectives on humanity's relationship with the gods and fate. The tragic effect described by Aristotle as supreme in the Greek classical drama is contingent on an audience understanding a story. So since Sophocles means to tell the story of Antigone's integrity and the cost of it, the backstory of the brothers is unnecessary and, for his audience, would be redundant. However, there is also artistic reason to start the story here. The lack of backstory and war drama keys us in immediately into the human dimensions of the story instead of making it an adventure story. It's a simple central question - what is the cost of integrity? - and by starting with Antigone's decision, Sophocles can make the dramatic stakes solely contingent on this.
