Learn Italian: Ce ne mancano cinque per l'imballaggio. - We are missing five of them for the packaging.

Because mancare works like piacere. The English subject ("we") becomes the indirect object in Italian ("to us"), and the thing that is missing becomes the grammatical subject. Here, the subject is the plural quantity cinque (five), so the verb must be the plural mancano.

Where does the "we" come from in the Italian sentence?

It comes from Ce, which is the indirect object pronoun ci (to us) in disguise. When ci is placed right before the pronoun ne, its vowel changes to an "e". So, Ce ne mancano cinque literally means "Five of them are missing to us."

Why do we need the pronoun ne here? Could we just say Ci mancano cinque?

No, you cannot drop ne in this context. Whenever you state a number or quantity (like cinque) but omit the actual noun you are counting, Italian grammatically requires ne (of them) to replace that noun. While English allows "We are missing five", Italian demands "We are missing five of them".

Can mancare also mean "to miss" a person, or just "to lack" an object?

It means both, and it uses the exact same "backward" structure for both! If you want to say "I miss you", you say Mi manchi (literally, "you are missing to me"). In this sentence it happens to refer to a physical lack, but the grammatical rules are identical.