Мы должны были уехать в июле, но из‑за работы остались в городе.

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Breakdown of Мы должны были уехать в июле, но из‑за работы остались в городе.

• город: the city
• быть: to be
• работа: the work

Questions & Answers about Мы должны были уехать в июле, но из‑за работы остались в городе.

What exactly does должны были mean here, and why is it in the past tense?

Должны были is the past-tense form of должны (from должен = “must / have to / should”).

• мы должны были уехать ≈ “we were supposed to leave / we had to leave.”
• The auxiliary были puts the obligation into the past: they previously had that obligation or plan.

So the structure is:

• мы должны были
  • infinitive (уехать)
    “we were supposed (in the past) to leave.”

It implies a planned or expected action in the past, which in this sentence did not happen because of the second part: …но из-за работы остались в городе.

Why is it должны были уехать and not something like надо было уехать? What’s the difference?

Both are possible, but they have slightly different nuances:

• должны были уехать
  – “were supposed to / were obligated to leave.”
  – Sounds a bit more like a duty or strong expectation, sometimes tied to a plan, schedule, arrangement.

• надо было уехать
  – “should have left / it was necessary to leave.”
  – More about practical necessity or hindsight (“we really should have left”).

In this sentence, должны были fits well because it suggests there was a concrete plan or arrangement to leave in July (maybe tickets, reservations, agreed dates, etc.), which got canceled because of work.