Learn Russian: Из-за того что шёл дождь, мы остались дома. - Because it was raining, we stayed at home.

While потому что is a neutral 'because', из-за is specifically used for external causes, often with a negative tone (meaning 'due to' or 'on account of'). The rain was an obstacle that ruined the original plans. Because из-за is a preposition that requires the Genitive case, we use the pronoun то in the Genitive (того) to connect it to the rest of the phrase.

Literally, what does шёл дождь mean?

It literally translates to 'the rain walked'. In Russian, precipitation 'walks' using the verb идти (to go on foot). We use the past tense masculine form шёл because the noun дождь (rain) is masculine.

Why is the verb остались perfective here, while шёл is imperfective?

The verbs show two different types of actions. шёл is imperfective because the rain was an ongoing, continuous process in the background. остались is perfective because staying home was the completed result or final decision made due to that rain.

Why does the sentence use дома instead of домой for 'at home'?

дома refers to a static location ('at home'), which fits perfectly with the verb остались (stayed). домой indicates direction or motion towards home ('homewards'), which you would use with verbs like going or returning.