んだ is a colloquial contraction of のだ. This is definition #2 of の that you can see here.
https://jisho.org/search/%E3%81%AE
It turns what came before it into a noun or noun phrase.
You can see the Jisho entry for the various ways of saying のだ here.
https://jisho.org/word/%E3%81%AE%E3%81%A7%E3%81%99
It has a variety of uses, but it is often used in an explanatory sense. That's just the simplest of explanations though.
のだ and だ are also often placed at the end of a sentence in Anime Japanese. this then becomes shortened to んだ. there is no real reason for this and it is recommended to not use anime Japanese in real life.
It's true that だ alone will often stronger in intensity than people imagine it to be, because they think they should just use it wherever they would have used です if it was a polite sentence.
However, there are plenty of times when it's appropriate to use だ, のだ, and んだ in everyday Japanese. To suggest it's not common in everyday speech is a little strange to me.
As Leebo said, completely normal in everyday Japanese, though I'd suggest that のだ is used only in the most formal situations.
As for the original question, I'd guess that it's めいれいけい, a command, as this form is sometimes used in place of the standard form (逃げろ).
Picking Apart "Nigerundayo" 『逃げるんだよ』
I'm sure a lot of you are familiar with the meme from Jojo's Bizarre Adventure. "Nigerundayo", said by Joseph Joestar.
I know that 逃げる means to escape or get away. And I know that だよ is shorthand for ですよ. But I'm having trouble with the ん in the middle. Where does that come from?