Learn Polish: Cholera, znowu pada! - Damn, it is raining again!

Yes, literally it means the disease cholera. However, in Polish, it is incredibly common to use cholera as an exclamation similar to "damn!", "crap!", or "hell!". It is considered a mild swear word, so you should avoid it in formal situations, but you will hear it everywhere in everyday life.

Why is there no word for "it" before pada?

In Polish, weather verbs like pada (it is raining) or wieje (it is windy) do not need a dummy subject like "it" in English. The verb simply stands alone in the third-person singular form.

Does pada specifically mean "raining"?

The verb padać literally means "to fall". When used alone about the weather, pada usually means "it is raining", though in winter it often implies "it is snowing". If you want to be specific, you can say pada deszcz (rain is falling) or pada śnieg (snow is falling).

I sometimes see znów instead of znowu. Is there a difference?

No, they mean exactly the same thing: "again". Znowu and znów are completely interchangeable, though znowu is slightly more common in casual spoken Polish.