Being jealous is what's called a free adjunct. It would be very difficult to explain exactly what linguists understand about it to you, since it requires a lot of technical knowledge.
The best way to explain it is this. Let's start with the sentence:
Mona didn't let her boyfriend dance with any of the cheerleaders.
At this point, the listeners doesn't know why Mona didn't let her boyfriend dance with any of the cheerleaders. However, the speaker wants to let the listener know why Mona didn't let him, so they want to add that Mona was jealous.
The most straight-forward solution would be your sentence,
Mona didn't let her boyfriend dance with any of the cheerleaders because she was jealous.
The reverse also works (Because...Mona...). But they require two clauses inside the main sentence.
[[Mona didn't let her boyfriend dance with any of the cheerleaders] [because she was jealous.]]
Instead of having to put two clauses inside the main sentence, why not just have one with some extra information tacked on?
[Being jealous,] [[Mona didn't let her boyfriend dance with any of the cheerleaders.]]
This is analogous to:
Jealously, Mona didn't let her boyfriend dance with any of the cheerleaders.
The difference is that with jealously, it moved from the main clause to outside it. In other words, the original sentence was:
Mona jealously didn't let her boyfriend dance with any of the cheerleaders.
And then the jealously moved in a process we call topicalization. Topicalization in English moves parts of the sentence to the front so they're more prominent.
↓----------¬
Mona jealously didn't let her boyfriend dance with any of the cheerleaders.
With being jealous, however, it didn't move from anywhere. The speaker just thought of it and attached it to a point where it's allowed, which includes the front of the sentence.
Being jealous
↓
Mona didn't let her boyfriend dance with any of the cheerleaders.
In conclusion, being jealous is basically a quick way of saying "Mona was jealous, therefore...." with fewer words.