The main quest to unlock the 3rd phase of the Shattered Sun Offensive is quick and easy. You'll report to Harbinger Inuuro at the new building. He will ask you to go to the Dawning Square and kill 6 Burning Legion Demons, and drive the provided Shattered Sun Banner through the corpse of the Emissary of Hate, who will come after enough demons are killed.
To do the quest, you'll want to head south of the Sanctum to the Dawning Square. If you'll recall, you came to this same square for The Missing Magistrix quest (47,44 for those of you with location mods). Kill any six of the demons walking around here or in the nearby buildings, be they Shivan, Infernal, or Hellion, and you'll get your kill credit. Once the Emissary of Hate spawns, kill him and use your banner while targeting him. You'll drive your banner through his corpse and receive your quest credit. It should be noted that, as of this writing, you don't have to get angry if someone else grabs the Emissary before you do. Once he is dead, anyone can drive the banner through his corpse and get quest credit, and multiple banners can be used on his corpse, so feel free to plant your banner even if someone else did the killing.
Once you've done this, you can head back to the sanctum and turn in the quest. You'll get 10 gold and 10 silver, and will help your server unlock the next step of the Offensive. Once this phase of the offensive is over, this quest will be renamed "The Battle Must Go On," but the objectives and the method of completing them will remain the same. For more information on this quest, you can also check out Wowwiki or Wowhead.