Jung typed himself as Ti-Ne and Freud as Fi-Ne, so those examples aren't the greatest I guess... I have great difficulty understanding how Ni works though, as soon as I think I understand it something new makes me doubt that understanding...
Jung typed himself as ISTP too. Keirsey identifies him as an INFJ. There's a lack of tangibility with Jung that makes me think he was INFJ. INFJs are Ni-doms like INTJs, but they lack Te, which is oriented towards the exterior and objects which makes them more empirical and lucid than INFJs. INFJs are oriented towards the metaphysical, while INTJs are oriented towards the physical, generally.
Jung identified Freud as an ESTJ as well. Freud was quite the opposite of feeler, or at least empathic. The foremost reason Jung and Freud split was because Freud couldn't accept Jung's dissent in thoughts as his pupil; Freud was a intellectual dictator and possibly a narcissist. Quite contrary to those with dom/aux Fi, Freud saw no value in art, which is typical of Te users. Freud is probably xNTJ. Some describe him as an introvert, others as an extrovert.
"I have found little that is "good" about human beings on the whole. In my experience most of them are trash, no matter whether they publicly subscribe to this or that ethical doctrine or to none at all. That is something that you cannot say aloud, or perhaps even think." - Freud
"The liberty of the individual is no gift of civilization. It was greatest before there was any civilization." - Freud
"Men are strong so long as they represent a strong idea they become powerless when they oppose it." - Freud
"Sadism is all right in its place, but it should be directed to proper ends." - Freud
However, whether Jung and Freud possessed Ni or not, and they very likely did, this is all irrelevant to the topic, which is the Ni insight into the human psyche.
My understanding of Ni is as follows:
Ni is quite abstract-oriented and often deals with will-power, the force of the will of the Ni user fuels the Ni and directs the Ni-dom towards a few goals with strong determination. They are often very ideological. Marx, Nietzsche, Rand and Hitler, for instance. They often understand their own ideas as being indisputably in the right.
The other aspect of it is its insight, which I touched upon in the OP. Ni users tend to get insight, basically eureka moments, often seemingly out of nothing. Ni is also oriented towards the future. They're quite visionary. However, because it is the opposite of Se, it rarely deals with that which is tangible. One could say that whereas Se-doms can't see anything that isn't planted in immediate reality, Ni-doms don't notice things right in front of them (and sometimes reality itself
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). From this we have that they read between the lines, and often like to interpret, which at times can be to the chagrin of objectivity, because they interpret and channel information to the benefit of their ideas and convictions.
As mentioned, sudden insights is part of it. One benefit I have of it is that I can gather a small foundation of information on something, a small piece of the body, and I gain an intuitive understanding of the whole, just from that particular. Basically, just as I enter the dark tunnel, I arrive at the other end to see the light at the end. It fills in blanks, but it also excludes possibilities. Whereas Ne is expansive, Ni is contractive. Ne is an explosion of ideas, Ni is an implosion of ideas.
Ni also deals quite heavily in literary techniques due to abstract-mindedness. Metaphor, parable, allegory... you name it, anything. That's why they're seen as gifted orators, often as prophetic.