Learn Spanish (Latin America): Ella mete el libro en la caja.

Both verbs mean "to put," but meter specifically means to put something inside or into an enclosed space, like a box, bag, or drawer. Poner is a more general "to put" or "to place." Because she is putting the book inside a box, mete is the perfect fit.

Since this lesson is about complex prepositions, could I use dentro de here instead of en?

Yes! En is a simple preposition that works perfectly to mean "in." However, if you want to be very specific and emphasize that the book is inside the box, you could say Ella mete el libro dentro de la caja.

If en can mean "in", "on", or "at", how do we know it means "in" here?

Context gives it away! The verb meter strongly implies putting something inside, and a box (la caja) is a container. If the sentence used a flat surface, like Ella pone el libro en la mesa (She puts the book on the table), en would mean "on."

Are la caja (the box) and el cajón (the drawer) related?

Yes, they share the same root! Just remember that la caja is a standalone box, while el cajón is a drawer built into a piece of furniture. Both are great places to put (meter) things!