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

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