Learn Italian: Loro mettono le scarpe nuove in una scatola.

It comes from the regular verb mettere (to put). To make the "they" form for regular -ere verbs, you drop the -ere ending and add -ono, which gives you mettono.

Why does the word for new (nuove) come after the word for shoes (scarpe)?

In Italian, descriptive adjectives usually go directly after the noun they describe. Also notice that nuove ends in -e to match the feminine plural noun le scarpe.

Why do we use in instead of a combined word like nella here?

Because the sentence says "in A box" (in una scatola). You only combine the preposition in with definite articles (the words for "the") to make combined words like nella or nel. With indefinite articles like una, they stay separated.

Is the word Loro (they) strictly necessary here?

No, you could just say Mettono le scarpe nuove in una scatola. Because the verb ending -ono already tells us the subject is "they," the word Loro is optional and mostly used for emphasis or clarity.