Basically you do a mission with the followers of your classhall and they give you an object that starts a quest that sends you to kill a boss inside the Antorus raid.
The real question is
which kind or gearlevel drops that chestreward when you complete the quest
Well, from default is raidfinder level gear.
If you did at least 12 bosskills in normal raid runs, then your chest will reward you with raidnormal level gear
If you did at least 12 bosskills in heroic raid runs, then your chest will reward you with raidheroic level gear
Note: You need those 12kills done before you start/complete the mission on your classhall.
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Please forgive my lack of a scientific approach, I am not sure how many bosses I had killed before I picked this up (re: Revocxam's comment), but I got normal gear from my turn-in. Here is what I do know:
I didn't clear Normal or Heroic last week
I only have one kill this week on normal (
on the character who turned in the quest, more kills on other characters, though - not sure if that matters)
According to raider.io, the character in question has 7 normal kills, presumably (afaik) most of which were indeed killed before I got this quest
All in all, I am not sure if there is account math or something, or if the baseline is norm gear and doing lots of norm kills gets you heroic and lots of heroic can get you mythic perhaps. I didn't run these quests very many times prior, and always just kind of let fate decide (ie not planning out when to pickup, turn in, etc).
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?