It's a worldview. Everything is understood in terms of cause and effect, as opposed to what things are and what kind of thing is something (archetypes, Si). Implied in "cause and effect" are all the dynamics to which everything is subject. It ends up being classified as "intuition" and "mystical" because it isn't concrete the way archetypes of things are concrete. One is stuck waving one's hands and saying, "isn't it obvious that it works that way?" and wondering why it isn't obvious to everyone else.
The intuition can seem even more mystical and magical because thinking in terms of "cause and effect" makes it obvious when cause and effect are inconsistent: think of Sherlock Holmes mysteries, where the protagonist simply knows that something is true or untrue because the supposed causes are missing effects, or effects are present that imply unmentioned causes.
For an ENFP, Ne is the leading function. It deals more with out-of-context thinking - finding relationships between ideas in different contexts/situations. Ni necessarily explores within a single context at a time: it discerns which causes/effect belong in a context. Ni isn't concerned with the "things" in the context: a shop isn't identified by its name, but rather by what it sells, and more specifically, in terms of what the Ni dom would consider buying from it.
So if you think of "cause and effect" as worldlines, as strings, Ni sees these flows naturally, without having to think hard. Ni memorizes how these flows flow. Then in spite of having a "bad memory for details" (in Si terms), if you give an Ni dom a few pieces of information, Ni immediately starts seeing how the pieces of information flow and fit (or don't fit) with each other. If Ni already has a lot of stored information (of flows and a few specific details), new pieces of information immediately fit or don't fit, producing the intuitive insight. The new information has to align with the current flows: if it doesn't fit, it's wrong (obviously so, to Ni, which can make for difficult arguments and accusations of stubbornness), and if it does fit, it immediately points to new truths that aren't at all obvious to those who aren't thinking of the world in terms of these flows.
Also, because they are flows, cause and effect, many of the insights of Ni tend to be remarkably accurate predictions about what will happen in the future. Just as often, like in the Sherlock Holmes example, they are remarkably accurate troubleshooting, seeing the cause that must have occurred (where to Ti, the logic for such a conclusion is rather incomplete).
Anyway, that's pretty much it. All of the mystery of Ni just has to do with its wordlessness. We have no words for the flows we see, and those flows are so obvious to Ni, it doesn't occur to Ni doms that no one else sees them, thus Ni self-descriptions of thought processes tend to be, um, rather poor and not very useful descriptions.
