Learn Italian: Quando il magazzino è vuoto, ci mancano i prodotti. - When the warehouse is empty, we lack products.

This is because mancare works backwards compared to English, much like the verb piacere. In Italian, you do not lack a product; rather, a product is lacking to you. Here, i prodotti is the grammatical subject. Since it is plural, the verb must be plural: mancano.

If i prodotti is the subject, how do we express the we part of we lack products?

We use the indirect object pronoun ci (to us). The literal translation of ci mancano i prodotti is the products are lacking to us.

Why is there a definite article before prodotti (i prodotti) when English just says products?

Because i prodotti is the grammatical subject of the sentence. In Italian, subjects almost always require a definite article, even when speaking about a broad category of things in a general sense.

Is magazzino a false friend? It looks a lot like magazine.

Yes, it is! magazzino means warehouse, storehouse, or storage room. If you want to talk about a magazine you read, you would use the word rivista.