Quests are special objectives which you need to complete in order to unlock the next sector of the building.
Building sectors are split in the set of 5 rigs and 1 cooling system, for those 5 rigs.
There are multiple quests per sector, usually tied to currency unlocking, mining and trading. For that reason, it's highly encouraged to unlock and mine the best currency you can at the moment, since that will help you with both the building income and resolving the quests. Also worth mentioning is that you should always upgrade your currencies, since that speeds up the time needed to finish quests, since the currency will be worth more.
Once you've finished the quest (you can track the progress on the progress bar for the current quest) , in order to complete it, you need to claim the reward. Once you've claimed all the rewards, the unlock button's going to get activated, and once pressed, you'll start unlocking the next sector of the building. Once finished you will get rewards from the quests chest.