Learn Croatian: Pada kiša i zato sam doma. - It is raining and that's why I am at home.

Croatian doesn't use a placeholder word like "it" for weather. Instead, you simply say the weather event is happening. For rain and snow, the verb padati (to fall) is used, so Pada kiša literally translates to "Rain falls."

I always mix up zato and zato što. How do I know which one to use here?

Zato means "that's why" or "therefore" (showing a result), while zato što means "because" (showing a cause). Since the first part of the sentence gives the cause (it is raining) and the second part gives the result (you are at home), you use zato.

Could I say u kući instead of doma?

You could, but it changes the focus of the sentence. U kući specifically means "inside the house" (referring to the physical building), whereas doma is an adverb that naturally means "at home."

Why is the verb sam placed right after zato?

Sam is an unstressed word (a clitic) that strongly belongs in the second position of its clause. After the connecting word i (and), the new clause starts with zato in the first position, which forces sam right into that second spot.