Die in this case is playing the role of a demonstrative pronoun, similar to "this" or "that" in English. It's odd for an English speaker to see a word that's normally seen as an article used this way, but you can think of it as a different word that happens to look and sound the same as the article. As English has a "this/these" set and a "that/those" set of such pronouns to account for meaning and inflection, German has a der/das/die set, a dieser/dieses/diese set, and a jener/jenes/jene set, though the last set is disappearing from the spoken language. German inflects by case as well as gender and number so each of these sets has an inflection table with 16 entries, though they all follow the same general pattern. Also, the circumstances for when to use which set, as well as when to use any of them in place of a personal pronoun, are rather tricky to master, but I'm sure the rules have been covered elsewhere on this site as well in any good German grammar, so I won't try to explain them here too.
The other issue here is flexible word order in German. Remember that any functional part of a sentence can go before the verb, not just the subject as in English. You can tell that wir is the subject in Die brauchen wir because it's wir and not uns. Putting die first marks it as what the sentence is actually talking about even if it's not the grammatical subject. Note that "Those we need," would be possible in some varieties of American English, due no doubt to the influence of the many immigrants who spoke German or some dialect of German such as Yiddish.