Mike Tom Hanks observes his two sons fighting, with one insisting that a promise doesn't mean anything. To help them understand that a promise does mean something, he tells them the story of his youth. In 1969, 11-year-old Mike Elijah Wood and his 8-year-old brother Bobby Joseph Mazzello, their mother Mary Lorraine Bracco, and their German Shepherd Shane relocate from New Jersey to Novato, California after their father leaves them. There, Mary weds a new man named Jack MacKenzie, whom the children call The King Adam Baldwin. Unbeknownst to Mary, The King is an alcoholic who often gets drunk and beats Bobby. The King also repeatedly plays Hank Williams's Jambalaya (On the Bayou) on his record player.
Seeing that Mary has found happiness at last with The King, Bobby swears Mike to secrecy about the abuse. Instead, the two boys seek adventures to occupy the time that would otherwise be spent with The King; they recount the “seven great abilities and fascinations” of childhood while exploring their new surroundings and dealing with the neighborhood bullies. An unsupervised incident in the kitchen in which the boys attempt to create an anti-monster potion to get rid of The King leads to Bobby being hospitalized by The King, but Shane gets revenge on The King by violently biting him on the arm. After spending time in jail, The King is released following the death of his mother and returns to their house with flowers and chocolates promising never to drink or abuse Bobby again; he violates his promise while the boys are at school and nearly kills Shane. As a result, the brothers devise a plan for Bobby to escape The King once and for all. Mary also starts to catch on to The King's true nature and finally requests a divorce.
Inspired by the urban legend of a boy named Fisher Sean Baca who attempted to fly away on his bicycle over a cliff nicknamed “The Wishing Spot”, the two convert their eponymous Radio Flyer toy wagon into an airplane in the hopes of sending Bobby and Shane away from harm. They draw up a schematic diagram with wings and an engine and scavenge numerous parts, secretly using The King's tools to build the aircraft in their shed. The boys also raise money through various means, such as retrieving lost balls on a golf course and selling them back to the golfers. After leaving a farewell letter for their mother, the brothers take the device to the cliff at night, but The King discovers their plan and attempts to stop them, prompting Shane to furiously attack him. Bobby then speeds down the hill alone (knocking out The King with the wing of his plane) and triumphantly soars into the air as Mike and Shane look on. Mary arrives with police officer Jim Daugherty John Heard who arrests The King. Though Mike never sees Bobby again, he continues to receive postcards from him from places all over the world.
Back in the present, Mike reiterates to his sons the importance of keeping a promise, and imparts a lesson about history being in the mind of the teller. He concludes his story by saying, "That’s how I remember it."
