This is best understood by thinking of it as a form of ので. The reasoning becomes clearer if we make the sentence complete.
月曜日は試験があるので、飲み会に行けない。
I’ve got an exam on Monday so I can’t go.
近所の人が夜間を通して音楽をかけたので、今日は眠い。
Today I’m tired because my neighbour played music all night long.
アナ
眠そうだね。
Anna
You look sleepy.
肇
近所の人は夜間を通して音楽をかけたんだ。
Hajime
My neighbour played music all night long.
It is also used at the beginning of explanations…
宿題をしたんですが、持ってくるのを忘れました。
I did my homework but forgot to bring it.
彼に電話したんだけど、出なかった。
I called him but he didn’t pick up.
Finally, one extremely useful phrase: ending the sentence with んですが when we want to make a request.
この手紙をイギリスに送りたいんですが。
I want to send this letter to the UK.
ちょっと聞きたいことがあるんですが。
I have something I want to ask you quickly.
We don’t really have a direct equivalent in English (it’s kind of like verbally expressing “…” at the end of the sentence as a prod for the listener to take some action for you). In the above examples we want someone to go ahead and say, “Sure, put it on the scales” or “No problem. Go ahead”. The んですが expression is simply a way to omit the obvious “… are you able to do that?” or “… is that okay?”
2. For Emphasis
何してるんだ？！
What on earth are you doing?!
それは嘘なんだ！
That’s a lie!
Remember that with nouns and na-adjectives we need a な before the のだ.
3. To Express What Should Be
何をやっても、ベストを尽くすのだ。
You should do your best.
言い訳するんじゃないよ！
You shouldn’t be making excuses.
マイク
昨日の夜は帰りに自転車から落ちて怪我したんだよ。
Mike
Last night I fell off my bike on the way home and hurt myself.
沙織
お酒を飲むんじゃなかったね。
Saori
You shouldn’t have drunk alcohol.
Note how in the first sentence the んだ is used for emphasis whilst in the second it is used by the speaker to express “should not have”.