First, let’s start with some background.

At the Seder, before we begin to tell the story of the Exodus, we take the middle of the three matzahs that are stacked under the Seder plate and split it into two parts. We return the smaller half to its place between the other two matzahs, and we place the larger half in a bag, or wrap it in a cloth, and then set it aside. The matzah that is set aside is called the afikoman, and it is eaten for “dessert” after the Seder meal in commemoration of the paschal sacrifice.(The word afikoman is from the Greek epikomen or epikomion, meaning “that which comes after.” Others say that it comes from the Aramaic afiku min, which means “bring out various delicacies with which to end the meal.”)

Now to the question of why we hide the afikoman:

The simple reason that we put the afikoman aside or hide it, is because we will eat this matzah only near the very end of the Seder, and we don’t want it to get mixed up with the other matzahs at the table.1

In addition, when putting aside the afikoman matzah, the custom is to wrap it in some sort of cloth or napkin as a remembrance of the way the Jews left Egypt with their soon-to-be matzahs,2 as described in the Torah:

The people picked up their dough when it was not yet leavened, their leftovers bound in their garments on their shoulders.3

Some have the custom to hide the piece of matzah that was set aside for the afikoman, and have the children find it and then return it only in lieu of a promised gift. This custom is based on a statement in the Talmud: “We snatch matzahs on the night of Passover in order that the children should not fall asleep.”4 In other words, the game of hiding the afikoman and the accompanying bargaining for a gift is an activity to engage the kids and make sure that they don’t fall asleep during what is invariably a long evening.

Others do not have the custom for the children to take or “steal” the afikoman, for fear that the activity would give them a taste of stealing.5

On a deeper level:

The afikoman represents our liberation from Egyptian exile. That redemption, however, was not a complete one, as we are still awaiting the final redemption with the coming of Moshiach. Setting aside or hiding the larger half of the matzah reminds us that the best, the real redemption, is yet to come, still hidden in the future.6

For additional afikoman customs, see Afikoman Customs.