Chekhov's gun is a writing principle that states that everything in a story should be there for a reason. Chekhov's famous example is that if we're told a rifle is hanging on the wall in chapter one, then someone needs to fire it in chapter two, otherwise why tell us there's a rifle there? Chekhov tells us not to waste time with details that aren't important. Talk about only what is in service to the story, no matter how irrelevant it may seem at the start.
Another Year Wiser: Orbital Operations for 19 February 2023
Chekhov's Gun is about lots of things. First and foremost, yes, as the quote says, it's about narrative economy – you often only have so much space and attention to play with, so, as Strunk and White might have it, 'omit needless words'.
In fact, specific sensibilities when it comes to narrative economy vary between mediums, commercial intent, and even just writers and their own voices. I think you've got a lot more latitude in your average novel than your average comic or TV show. That's partly because the latter two, at least in the commercial shapes we tend to talk about most often, have specific, formalist constraints (page count, episode length).
It's also down to the dependencies of including an element. A prose writer can just toss out a bunch of details without any marginal cost or wider implications. In comics, someone's got to draw that. In TV, it might mean costuming, propmaking, casting. In video games, concepting, modelling, rigging, animating, capturing, or all manner of thing, depending on what you're asking for.
This is where narrative economy collides with actual economy, which is very important.
(The same can be said even of prose or 'cheaper', marginally speaking, forms of writing. Even if I can produce an arbitrary amount of prose without issue, I may have a limited time assigned for it given what makes financial sense for that piece of work to look like. Even if I can write faster than your average bear, it might be incumbent on me not to do too much if there are specific formalist expectations of the content, or even just because then it becomes time intensive to edit or QA.)
But there are some other useful aspects to Chekov's Gun. The inverse of its formulation, for instance: if you need to have someone fire a rifle in chapter two, you need to show that there is a rifle in chapter one.
This is, in some ways, 'earning' that moment (though there is more breadth to what 'earning a beat' can be). But also it's just about making the story feel intentional and inevitable. Often, it can be surprising the audience ('a character just pulled a gun!'), but making that surprise feel concrete -- resting on information that was available to them ('of course! the gun was on the wall the whole time!'). Though outside of Chekov's Gun, this can be earned in softer ways ('of course that character carries a gun!').
I'm reminded also of Sanderson's Law of Magic, which is interesting even if I have mixed feelings about it. Paraphrasing from memory: the level to which you are allowed to use a magic system in a work of fantasy to solve plot problems (e.g. a character using magic to get out of a fix) is proportional to the level of hard, established rules of that system. It's fine to have handwavy magic that behaves in ways that are inconsistent or hard for the audience to pin down, but if you then use them to get characters out of trouble, you're creating a situation where you're not bringing the audience along with you. It becomes effectively deus ex machina, or even 'Sonic Screwdriver Syndrome'.
(I'm increasing levels of tangent deep now, but one of the things that made me care less for Doctor Who than I already did was watching a two-parter where the first episode ended dramatically with the Doctor et al. surrounded by approaching Cybermen. Oh no! How will they escape?! And then the second episode begins with the Doctor whipping out the screwdriver and doing... something. I'm not even sure what. It felt like the most baseless manufactured tension and a waste of everybody's time.)
Anyway, Chekov's Gun here is partly about explaining at leisure what will be dramatically important in haste later.
Last thing on Chekov's Gun: I think it's also paying off the promise of what you're putting out there. Not just in the strict narrative economy sense, but: if you're explaning some cool facet of the world or something about a character that seems specifically interesting... you're making a kind of promise to the audience that they'll see that play out somehow. Not because that's an unnecessary detail, but because you're getting them excited about something that sounds cool, and it's better to pay that off than not much of the time.
But above all, as the quote says, this is a principle, not a prescription. I can imagine a reductive, CinemaSins-esque analysis of a work that points to all the details that it includes but doesn't pay off (or that it pays off without setting up). Which is not... how this works. This is a shorthand description of what tends to make stories connect up in exciting ways, not the be-all and end-all of how to make them.