Visiting the Mystery Box Shop is accomplished by tapping the MBS Quest in the Quest Box, then when it opens tap the Gift Package on the bottom right corner -- which will take you to the MBS. You will also be able to see what the current quest or Task is, and what your overall progress is for this week's MBS Key!
The MBS contains four prizes, each of which requires a larger number of keys to claim. These are:
Curiosity Box (1 Key)
This first level of MBS prize contains x3 Standard Prizes and x1 Common Items.
Loot Box (3 Keys)
This is the second prize, and it contains x4 Standard Prizes, x2 Premium Prizes, and x2 Common Prizes.
Bounty Box (5 Keys)
The third box contains x6 Standard Prizes, x4 Premium Prizes, and x1 Rare Item.
Fortune Box (10 Keys)
The fourth and final selection, the Fortune Box contains x10 Standard Prizes, x12 Premium Prizes, and x2 Rare Items.
The rub here is to get the best level prize -- the Fortune Box -- you will need to patiently work towards collecting Keys for 10 Weeks! Ah but clearly it is worth it!
The MBS Quests
As previously noted these span a wide variety and types of Tasks, from using a specific item in your Sim home to having X number of Sims employed at a specific career...
For the object quests you just need to make certain you have the required object in a Sim home and then have a Sim use it... For the career quests you will have to adjust the career paths for your Sims to meet the requirements.
You should consider the fact that a Sim who is already in a given career (even if they have mastered it) can quit that career and then re-start it and they will only lose one level of it -- in other words they will not start from zero. That being the case that may be the best way to deal with the requirements when it is just a single career.
When the requirement is for multiple Sims to join a specific career you will then need to make a judgement call on which get diverted to that career. It is a good idea though to have at least one Sim in each career, and just use the duplicates for the new career that the quest requires.
The Mystery Box Shop is a new quest-based bonus system. It uses a wide variety of quests as part of the weekly series, the ultimate goal of which is to unlock and obtain the key for each week -- these Keys are very important because they are the currency that is exchanged for the special prize boxes that can be obtained by visiting the Mystery Box Shop.