Learn Dutch: Het kind valt uit de boom. - The child falls out of the tree.

In English we use two words ("out" and "of"), but Dutch uses the single preposition uit. It means "out of" or "from" when referring to a movement out of an enclosed space or object (like the branches of a tree).

Could I say van de boom instead of uit de boom?

You could, but it changes the picture slightly. Uit emphasizes that the child was physically in the branches (inside the tree) and fell out of them. If you use van (from/off), it sounds more like they simply fell off the outer surface.

Why is the verb valt and not vallen?

The dictionary form of the verb is vallen (to fall). Because the subject here is het kind (the child), which acts like "he", "she", or "it" (third-person singular), we take the verb stem (val) and add a -t, giving us valt.

Why does the sentence use both het and de to mean "the"?

Dutch has two different words for "the". Kind (child) is a neuter noun, so it takes het. Boom (tree) is a common noun, so it takes de. You have to memorize the correct article whenever you learn a new noun!