Okay, let’s use this as an example.
On one level:
Firefly is a genre-bending blend of two standard forms, the Western and the Sci-Fi adventure. (And yes, I use the informal Sci-Fi deliberately, rather than the more purist SF.) That much is obvious. If you “get” that, then you “get” the basics of the show. And if the resulting hybrid format doesn’t particularly resonate for you, or if you just plain don’t like it, well, that’s fair.
On another level:
Firefly uses this hybrid form to examine the broad currents of history in a metaphorical manner. In any exploration of new territory, there will be a “core of civilization,” and a “frontier.” The forms these take will no doubt be particular to their time and place, but Firefly chooses to borrow familiar narrative archetypes to represent those two extremes. In the Firefly universe, “civilization” takes the form of a fairly cold but safe and ordered Star-Trek-ky world, while the “frontier” is literalized as the Old West, which is warm and human but messy and dangerous.
Placing these two extremes in opposition gives the storytellers an opportunity to illuminate, allegorically, the conflicts and clashes of our own real-world experience. This is also fairly obvious, though you don’t need to “get” it to enjoy the show. However, if you do, there is additional room for appreciation and enjoyment. Doesn’t mean it’s required that you do. Just means that a simple acknowledgement that such potential depth exists provides an opportunity for you to further engage with the show if you choose.
And on yet another level:
Firefly is a reconstruction of narrative itself. By choosing two very clear archetypal genres, Firefly inherits by implication their rules, their requirements and expectations. Heroes in the Star Trek universe make a lot of noise about being advanced, about being peaceful and civilized, but when their backs are against the wall they still pull out the phasers. Heroes in an Old West universe know they’re on their own, know the Law is a somewhat more fragile construct, and have a much shorter distance to travel from “civilized” behavior to what is, effectively, vigilantism.
Firefly pits not just the historical and social bases of the two cultural archetypes against one another but the narrative bases as well, which is to say, the mythologies. We, as viewers, expect the characters of each storytelling format to behave according to the rules of their stories. And Firefly, by mixing and matching two conflicting genres, finds the freedom to borrow, exploit, invert, and/or discard any of the standard tropes and cliches of the narrative forms in which it’s based.
By so doing, the show reflects back to us our own preconceptions, our own wishes and desires, and our own reasons for enjoying these storytelling forms. Sometimes the show gives us exactly what we want, and sometimes it undercuts our expectations. In “War Stories,” we get to see Mal rising, all alone, from the torture chamber, exactly the way a good hero should; but then a couple of minutes later, when he’s in trouble and Zoe says “this is something he needs to do himself,” Mal says, “no it ain’t,” and his friends step in to assist. A clearer example of the show’s subversive intentions, I think, can’t be found.
I think of Firefly as sort of a first draft of history: messy, cluttered, organic, occasionally contradictory, not always entirely satisfying from a standpoint of pure storytelling, but deliberately so; it’s what things are like before the mythologizers begin streamlining and simplifying the stories to make tidy morals out of them. It’s Star Trek with Kirk as kind of an imperialist jackass; it’s Rio Bravo with Dean Martin’s character at the center and John Wayne off to the side. This appeals greatly to me, because I like stories that allow me to think about the story-ness of them.
That’s something else to “get” about the show: its conscious attention to its own storytelling. If you’re as interested in narrative as I am, about the audience’s relationship to the story, about how people engage with protagonists and their objectives and what about certain stories we find satisfying and what we find frustrating, then there is a lot to love in Firefly. If you’re not interested in that, or worse if you find it boring or navel-gazing or pretentious, then… not so much with the love.
Now, all of this is not to justify Firefly specifically. It’s just an example, from my own perspective, since it has been mentioned more than once here. Really, the intent is to lead, somewhat circuitously, back to what RickJay* and others have pointed out: the ambiguity of what it means to say that someone “gets” something.
Does it mean “comprehend,” in the sense of intellectual understanding? I would argue, no, not entirely; that doesn’t capture the full connotation. One can grasp all three of the levels I laid out above for Firefly and yet still not enjoy the show overmuch, if one is not interested in what the show is trying to do.
Rather, to “get” something, one must not just understand the material, but connect with it, emotionally and intellectually, and sense, consciously or subconsciously, why the material is significant to oneself. As above, I am very, very interested in stories that allow me to consider, while I experience them, the construction of the narrative and the various moment-by-moment choices to either go with the genre expectations or turn them on their head. There’s lots of this in Firefly. I “get” that about it, because I’m predisposed to appreciate material that works on this level.
Which is why talking about “getting” something is such a vague and, as lissener says, potentially inflammatory term. It treads right on the line of being an insult, and, to my mind, is not a very useful foundation for a dialogue about a piece of material. Without a clearer understanding of all the connotations, without a deliberate effort to set aside the unwanted implications, a suggestion that “you don’t like it because you don’t get it” is, from my point of view, pretty much worse than useless, because it is so likely to be divisive, as observed by Aangelica and others.
I’d like to propose a moratorium on the term, at least when used in isolation. If it comes up, there needs to be a serious tangent to make sure everybody’s on board with what it means, before people start taking offense.
Oh, and just for the record, I don’t care for Moby-Dick either. 
