Learn Polish: One zostały tam miesiąc. - They feminine stayed there for a month.

The ending -ły is the non-virile plural past tense ending. In Polish, if a group doesn't contain any male people (like a group of only women, animals, or objects), you use the pronoun one and the verb ending -ły. If there was even one man in the group, we would use the virile pronoun oni and the ending -li, making it zostali.

Could we drop the word one and just say Zostały tam miesiąc?

Yes! Because the verb ending -ły already tells us that a non-virile "they" did the action, the pronoun one is optional. Polish speakers often include the pronoun only if they want to emphasize exactly who stayed there.

Why is there no Polish word for "for" in this sentence?

When you talk about how long an action lasted, Polish usually doesn't use a preposition. While in English we say "for a month", in Polish you simply state the time period directly as miesiąc.

Is the word miesiąc changing its case here?

Yes, but it is invisible! To express the duration of time, Polish uses the accusative case. Because miesiąc (month) is an inanimate masculine noun, its accusative form looks exactly the same as its basic dictionary form.