Learn Farsi: پسرها باشگاه بودند.

The verb is built from the past stem bud (was) and the personal ending -and, which is the standard suffix for "they" (third-person plural). Together, they make budand (they were).

Where is the Farsi word for "at" or "in" before bāshgāh (the gym)?

It is completely left out! In everyday Farsi, the preposition dar (in/at) is often dropped before places. It is perfectly natural to literally say "The boys gym were."

How do we know to translate pesar-hā as "the boys" instead of just "boys"?

Farsi does not have a direct word for "the." When a plural noun like pesar-hā is the subject of a sentence, it is usually understood from context to be a specific, known group, so we naturally add "the" in English.

Why is the verb budand at the very end of the sentence?

Farsi uses Subject-Object-Verb word order. Even for simple sentences describing where someone is, the verb almost always takes the final position.