Learn Italian: L'astuccio è vuoto. Dovranno comprare le matite. - The pencil case is empty. They must need to buy pencils.

This is the epistemic future! In Italian, you can use the future tense to make a guess, an assumption, or express a strong probability about something happening right now. While dovranno literally means "they will have to," in this context it means "they probably have to" or "they must need to."

Could I just say devono comprare (they have to buy) in the present tense instead?

Yes, but it changes the mood of the sentence. Devono comprare le matite is a stated fact: "They have to buy pencils." By using the future tense dovranno, the speaker is showing that they are making a deduction based on a clue (the empty pencil case).

Is astuccio exclusively used for school pencil cases?

Mostly, yes, it's the standard word for a school pencil case or pouch. However, it can also refer to other small protective cases, like a glasses case (astuccio per gli occhiali) or a small jewelry box.

Why is there an article before matite (le matite)? In English we just say "buy pencils."

Italian uses definite articles (like il, la, i, le) much more often than English, especially when referring to a class of objects as a whole. While comprare matite (without the article) isn't strictly wrong, Italian prefers comprare le matite to talk about getting the pencils that are supposed to go inside that empty case.