Why do children steal the Afikoman?
To show how much they love the mitzva. (Mekor Chaim - Chavos Yair)
Based on the Mishna in Pesachim that says the children grab the Matza so they shouldn't fall asleep. (Chasam Sofer)
Dogs watch the house from burglars. On the night of Pesach the dogs did not bark therefore they were vulnerable to thieves. As a Zecher to this we steal the Afikoman. (Michtav Sofer - Rav Shimon Sofer of Krakow)
Matza represents parnassa. The Ba'al HaBayis breaks it in half. He leaves half on the table representing Olam HaZeh and the other half he hides for Olam Haba. The little children who only know from Olam Hazeh try to grab the hidden half. The Ba'al HaBayis must watch it and make sure they don't steal it. (Ach Pri Tevua - Rav Tzvi Hirsch MiLiska)
When Eisav went in to get the brachos from Yitzchok, Yitzchok said "Ba Achicha BiMirma" you brother came with trickery. The Medrash Plia adds "and he took out the Afikoman." The brachos were given on Pesach. Therefore the children grab the afikoman to get the brachos. The brachos are the present that they ask their fathers to buy for them. (Rav Menashe Klein) Revach
To quote this article by Eliezer Brodt:
Where does this minhag come from? Many Acharonim point to the Gemara (Pesachim 109a), which mentions that we are “chotef” the matzos on the night of Pesach for the children. What does “chotef” mean in this context? The Rishonim offer different explanations. The Rambam (Hilkhot Hamets Umatsah 7:5) writes that on this night one has to make changes so that the children will notice and ask why this night is different, and one answers by explaining to them what happened. The Rambam adds that among the changes we make are giving out almonds and nuts, removing the table before we eat, and grabbing the matzah from each other. Rabbeinu Manoach points to the Tosefta as the source for the Rambam that “chotef” means to grab. Some Rishonim, quoting the Tosefta, write clearly that it means to steal.
It should be noted that this custom was far from universal and numerous rabbis opposed it including the Chazon Ish, Rav Shlomo Zalman Auerbach, and Rav Yossef Qafih. (ibid).
What is the source and significance of stealing the אפיקומן? It comes from Yackov stealing the ברכות from Eisav. It has a whole other level we tell the חכם a Halacha that אין מפטירין אחר הפסח אפיקומן why this Halacha? The answer illustrates the connection between Birchas Yitzchak and Pesach once more it was this Halacha that Yackov used when stealing the ברכות when Yackov brought the Meat from Korban Pesach Yitzchak was no longer able to eat because אין מפטירין אחר הפסח. We know that Yitzchak told Eisav “Go Prepare me food so I can eat and Bench You” .This means now that Yitzchak cant eat Eisav’s food he cant give him a ברכה..Then too throw in one last connection from Rabbi Yosef Chaim Sonnenfeld When Eisav came in and asked for his ברכה his father told him בא אחיך במרמה ויקח את ברכתך and if you calculate the Gematriah of במרמה it is exactly אפיקומן