Learn Hardcore Russian: Мне должны выплатить зарплату завтра утром. - They are supposed to pay me my salary tomorrow morning.

In Russian, with the meaning (someone) owes (something) to someone / (someone) is supposed to (do something) for someone, the person who is “owed to” is often put in the dative:

мне = to me So Мне должны выплатить зарплату literally works like To me they must/are obliged to pay (my) salary.

What does должны mean here, and why is it plural?

должны is the short-form predicate of должен (to be obliged / supposed to / expected to). It’s plural because the subject is implicit and understood as they (e.g., your employer, accounting, the company):

(Они) должны = They are supposed/required to If it were he or she, you’d use:
(Он) должен
(Она) должна If it were a single neutral “it” (e.g., the company as an entity), you’d often still use plural in real life if you mean people, but grammatically you could also see должно in other contexts.

Why is the subject not written? Who is doing the paying?