The description for the quest is a reference to Boba Fett, the bounty hunter who fell into a hole/sand worm and got eaten in the Star Wars movies; hence "Find a pit to throw him down".
You will receive a different level piece of gear from the chest reward to this quest depending on your progress (hence why there are 4 different quests ID's on wowhead for quests with the same name).
If you have cleared a difficulty level in previous raid tier you will automatically receive a quest for a chest one difficulty lower.
previously cleared mythic TOS, start off with a heroic Antorus cache
previously cleared heroic TOS, start off with a normal Antorus cache
I am unsure if this applies to clears after the new raid has dropped (it may require an AOTC kill).
It will also upgrade to a difficulty level you have achieved 12 boss kills on.
As far as I know they can all be kills of the first boss (no need for a full clear), however this will take 12 weeks as the boss kills must be eligible for loot to count.
Turn in is in your Class Hall.
Have you vanquished the enemy in Antorus?
Did you triumph in Antorus?