Like all great ideas, the notion of narrative necessity is really quite simple. At its most fundamental, it means quite literally that any story (as story) needs certain things to happen, otherwise there is no story worth telling.
People think in terms of narrative arcs. Our lives are narrative arcs (individuation paradigm); our school careers are narrative arcs; the political seasons and the 24 hour news cycle are full of narrative arcs. And these stories, in order to be told as stories, contain events required for the proper understanding of the tale.
When we look at the life story of a Hero, we see common elements. Whether the hero is Hercules or Harry Potter, these elements of the heroic paradigm are matters of narrative necessity --- they have to happen for there to be a story. When we look at history, we see certain patterns and matters of inevitability. Things need doing and those things need people to do them.
Narrative necessity, as regards writing stories and telling the histories of invented cultures and worlds, simply means we as writers or geopoets attune ourselves to the needs of the narrative and, well, "make up" those things that the story needs in order to be told.
One is the older meaning of narrative necessity: any of the events and situations required to ensure the consistency of the story.
In this view, a narrative necessity is an event which must happen, or a situation which must be present, in order to bring about a conclusion or a situation intended by the author; one could say that narrative necessities are links in the cause and effect chain which connects the premise with the conclusion. At the opposite end are events and descriptions which do nothing to advance the plot, but may be interesting and pleasant in their own right.
Another definition is a newer "post-modern" meaning of narrative necessity as a driving force which structures the story.
This newly identified kind of narrative necessity comes either from fundamental decisions about the story taken by the author, or, even more often, from undeclared (possibly unconscious) bias.
When I use the term in responding to querents here in Worldbuilding.SE, I mean most definitely the first definition. Most often, writers who ask questions here are asking for confirmation of whether some element in their story fits. Classic examples:
Vampire Who Still Eats Normal Food --- see Graham's answer
Excuse to Strand my Characters --- My answer to the related Meta question points out the fundamental narrative necessity of the original query. If you're writing a story set on a starship and you want your crew to become stranded, well, pick a scenario out of a hat. Oh, I know! Microfractures! The point about narrative necessity is that the actual event is not so important.
It could have been a software glitch or perhaps the USB plug that connects the navigator's thruster control to the worp drive came loose or it could have been a bunged up loo. (As a tip to "undeclared bias", I'd say that microfractures probably make for a better non-spoofy element of narrative necessity than a clogged toilet.)
Something needs to happen in order for the story to be a story, and that something is the narrative necessity.