Once the Warden receives this quest, a new location appears on the World Map, marked as Refugees. Travel to the location and fight the darkspawn, which consists of ten Hurlocks (including one Hurlock emissary) and three Genlocks. Once the fight is over (regardless of how many refugees survived) return to a Chantry representative for a reward.
Bug! Turning in multiple Chanter's Board quests at once will give you the reward for all of them, but only the last of them will show up on feedback. However, if you check your inventory, it will reflect that you get the full reward for each quest.