1. It shows that in times of war, human beings become objects. They have no face, no family, no history, and no importance to the other side. It is very much like a video game. There is no emotional ties to the person you are shooting. The main character isn't even given a name. He is just known as "the sniper". This is also shown by the "old woman, her head covered by a tattered shawl". She had done her damage, she had pointed him out. The other side had already been shooting at him, so they knew where he was. Her information was redundant, but he shot and killed her anyway --- after she had delivered her news. Note that it is not this death that he regrets.
2. War tears families apart. This story took place during the Irish Civil War. Just as in the American Civil War, brother was fighting against brother. They were fighting for ideals. The Irish Republican Army split over the Anglo-Irish treaty following the fight for Irish independence from Britain. There was the Provisional side that were pre-treaty and the anti-treaty group known as the Republicans. When he kills the enemy, he finds out that he shot his own brother.
3. Even in war, there is a basic respect for those who do their job well. The sniper doesn't care about the old lady, but when he shoots the enemy, watches him die, and then peers at his body, "he became bitten by remorse". He went to look at the body of his enemy because "He decided he was a good shot" He even dodges machine gun bullets to see who this person was. The man had been a worthwhile adversary.