んだ／のだ ／んです／のです is used when the sentence is concerned with another situation. Example son: 行<い>ってきます。I go out. mother: いってらっしゃい。Have a nice day. （The son came back home at once） mother: どうしたの。What happened? son: 雨<あめ>が降<ふ>ってきたんだ。It (has) rained.（So I came back home to take an umbrella with me.） ※んだ／のだ ／んです／のです expresses that 雨が降ってきた is concerned with my coming back home＝reason. When you noticed the rain, you can not say 雨が降ってきたんだ／のだ ／んです／のです. You should say あっ、雨が降ってきた／きました！. Oh, it (has) rained.

It's short for のです. I think it's when a speaker is explaining something or wants an explanation for something they are talking about or about an event currently is happening or has just happened.