The commandment to bring a korban Pesah says, “seh le-beit avot” (see Shemot 12:3), which appears to mean that a single sacrifice is brought for every family. Nevertheless, Rabbi Zeira teaches that this does not mean that all family members, including small children, are automatically included in the sacrifice by the Torah. Our Gemara points to a Mishna in Massekhet Pesahim that supports this ruling. The Mishna (Pesahim 89a) teaches that when a father tells his children “I will slaughter the korban Pesah on behalf of whoever gets to Jerusalem first,” the child who reaches the city first is credited with the sacrifice for himself, but his siblings are ultimately included as well. While it is clear from this Mishna that children are not automatically included, the Gemara questions how the other children can be included at all. The rule is that the only people who participate in a korban Pesah are those who had arranged to do so in advance. If the animal was already slaughtered for sacrifice, how can the children be included?
The Gemara responds that in this case the father was simply trying to encourage his children to be enthusiastic and hurry to perform the mitzva.
Tosafot and the Ran conclude from this discussion that children who are under bar and bat mitzva age are not obligated to participate in the korban Pesah, and none of the rules of the sacrifice apply to them. Thus, the rule that they need to arrange to be participants in this specific korban does not apply to them, and they can eat it even if they were not included.
Some point out that the Gemara in Pesahim offers an alternative explanation for the Mishna; it suggests that the children under discussion are adult children who are obligated in the sacrifice. According to that approach, the father would certainly have needed to include his children in the sacrifice before it was slaughtered, and that he, in fact, did so. The suggestion is that the father did not disclose this to his children however, and in the interest of encouraging their enthusiasm, made them think that only the child who arrived in Jerusalem first would merit participation in the korban.