Additional resources
The Young Adventurers’ Guides from D&D are fun introductions to the world of D&D–the monsters, classes, magic, etc. But they do not contain any rules. This guide is meant to present rules that can be used with those books.
The DnD Adventure Club publishes a simplified rules guide and a new adventure for kids each month. It’s meant to give kids the tools to run games themselves (as is this guide here).
Adventure with Muk is an activity book and introduction to D&D from D&D themselves. It doesn’t provide a lot of tools for running games but it can be a soft introduction for younger kids interested in D&D.