Saying "I'm sorry" can be one of the most difficult lessons to learn. Even as adults, we sometimes struggle with those two simple words, so it's reasonable for this to be just as tough as children, if not more so.
Children should learn to acknowledge when they've done something they shouldn't have or hurt someone, whether it was an accident or on purpose. Yom Kippur is a wonderful time to introduce the idea of saying "I'm sorry" to children; when children are not yet old enough to observe fasting on Yom Kippur, there are still ways to include them in observance of the holiday, such as focusing on the idea of repentance.
Books can be an easy and gentle way to begin a conversation with your child and provide kid-friendly examples of when and how to apologize. These stories explore saying "I'm sorry," and the characters in these books each have a different imeptus for needing to apologize - one is an accident, one comes from misbehaving, and one as a result of thoughtlessly hurting a friend's feelings.
For more about teaching children the importance of a genuine "I'm sorry," see this article from Kveller, which also highlights Tashlich at Turtle Rock.
Annie is excited about the Tashlich ceremony on the afternoon of Rosh Hashanah, when her family will walk to Turtle Rock Creek and throw crumbs into the water, as symbols of mistakes made the past year. As Annie leads her family through the woods stopping at favorite rocks, bridges, and waterfalls in her family's own Tashlich ritual, they think about the good and bad things that happened during the past year, and make plans for a sweeter new year.
by Jacqueline Jules, illustrated by Katherine Janus Kahn
The Ziz, a wonderful bird who lived long ago, is so big and clumsy he can't keep from bumping into things. When a tree he knocks over destroys the children's garden, he seeks God's help to fix things. "Bring me the hardest word," God intructs him, and the Ziz flies off to search. He brings back words like rhinocerous, rock, and Rumplestiltskin, but none is acceptable, until he makes an important discovery.
By Tilda Balsley and Ellen Fischer, illustrated by Tom Leigh
Brosh can't find his blue cap, and suspects one of his friends has taken it. When Grover returns the lost item, Brosh is glad that the High Holidays offer him a chance to say "I'm sorry."
by Sylvia A. Rouss, illustrated by Katherine Janus Kahn
When Josh breaks the rules and plays ball indoors, he finds himself apologizing not only to his parents, but to Sammy Spider as well.