Learn Italian: Lo scatolone è comodo e il gatto non vuole uscirne. - The box is comfortable, and the cat does not want to get out of there.

Here, ne is a locative pronoun meaning "from there" or "out of it" (referring to the box). When attaching a pronoun to an infinitive verb like uscire, you drop the final -e of the verb, giving us uscirne.

Could I put the "ne" somewhere else in this sentence?

Yes. Because volere is a modal verb (a helper verb expressing want, ability, or obligation), you have a choice. You can attach ne to the infinitive (non vuole uscirne) or place it right before the conjugated verb (non ne vuole uscire). Both are perfectly natural and mean the same thing.

Why is the word for box "scatolone" instead of "scatola", and why does it use "lo"?

Adding the suffix -one to scatola (box) makes it mean "large box" or "carton". Notice that this changes the gender: the feminine scatola becomes the masculine scatolone. Because it is masculine and starts with an s-consonant cluster (sc-), the definite article must be lo.

If I didn't use "ne", what preposition would I use to say "out of the box"?