んだ・んですExplanatory, Emphasis, The fact is

のです is a construction with several different forms, each meaning roughly the same thing. This expression behaves similarly to だ and です by themselves (asserting that something is true). However, the addition of の strengthens the relationship being highlighted. Due to this, it will come across a bit like 'it is a fact that (A)' in English.

In these sentences, we can think of the ん, or の as simply adding emphasis to the statement. Due to の adding emphasis, this expression will be heard very frequently in explanations.

Caution

There is no real difference between ん and の. Only の is considered 'correct Japanese', but ん evolved to make things easier to say. ん is the most common form in spoken Japanese, and the use of の could be considered a bit formal, or 'stiff'.

While both the だ and です forms are equally as natural, only the です form will be used if adding か to the end ～んですか. This construction shows that the speaker is looking for an explanation about something.