I decided to buy one since the main shops have changed. How do they work? Do I just keep it in my inventory or do I have to redeem it somewhere?

You need to keep it in your inventory, then it automatically discounts the price on items from the main shops. If you do happen to get the 15% discount buff, the discount card works second (so your 15% will come off the official price, then your 20 or 30% from the discount card)