The Gift Box temporarily holds items that are awarded to you from Shorty and Zeke!
You can access your Gift Box from the Festival home screen as long as there are items in it. It's the gold ribbon icon in the bottom center of your screen.
Keep in mind that the Gift Box is not meant to be used as storage, like your Shed. Each item held in your Gift Box must be claimed within a limited time. If unclaimed within this time, those items will be lost forever. Don't miss out!