Here's how it works with the Grab Bag:
Animal Crossing comes with a Memory Card. On it is on block's worth of data. If that Memory Card is used for an Animal Crossing town, the block of data will be used with the game. The person who starts the town will receive, after quitting and returning, a letter from Nintendo. Attached to the letter is a present. Using the A Button and Control Stick, remove the present from the letter, open the present to find a Grab Bag, and open the Grab Bag.
I assume that the Grab Bag will not open if you do not have enough empty spots in your inventory for the items it contains, but I cannot confirm this.