To be clear, I'm not looking to debate whether this is the best Trek film. Rather, I'm asking why so many people see it as such.
I enjoy TWoK well enough, and certainly it is a good film overall. But consider: it is much more militaristic than any Trek before and more than most Trek since, and relatively violent compared to TOS; there is no exploration of strange new worlds; tonally, it is quite different from most Trek stories. (To be clear, I'm not suggesting that these qualities are required for a "good" Trek film -- I'm just noting a few obvious ways that TWoK is unusual.)
In terms of TOS episodes, TWoK is probably most like a combination of "The City On The Edge Of Forever" and "Balance of Terror" -- which, to be fair, are beloved classic episodes, in part because they are somewhat exceptional compared to the rest of the series. So perhaps that gives us some clue as to why the film is so beloved.
In general, TWoK is ultimately about mortality. For all that the film professes to be about Khan, he really is just an Act of God (in the natural disaster sense), creating an unstoppable force that Kirk must humble himself against. The film is really about Kirk learning to confront death -- heightened by the contrast of the new life of Genesis and in his newly-rediscovered son. And that is something that the film did which was new: able to plumb the depths of Kirk's emotional journey at greater length thanks to the larger screen and the longer format.
But, again... it's a great film, but I don't know that it's obvious to me that Kirk learning to deal with the no-win scenario particularly epitomizes what "Star Trek" is (whatever the hell Star Trek actually "is"). In that respect, The Voyage Home seems like the most obvious candidate -- whatever Star Trek "is", to me TVH "feels" more like it than does The Wrath of Khan.
So, why has TWoK earned such a place of acclaim?
(PS: I could write a similar post about First Contact, whose popularity also confuses me.)
I saw The Wrath of Khan as a kid, quite possibly before seeing any of the series, so there's never been any question of it not being representative of Star Trek for me, though I can see how someone approaching chronologically might see more of a disparity.
However, TOS had plenty of deaths, including destruction of starships, as did V'Ger's, uh, collection of data, so does TWoK really stand apart in that regard? Chekov and Terrell kick off the plot while surveying planets, encountering a strange alien creature, and Kirk and co. find an underground paradise; I see that as fitting the explorative aspects of the show, at least somewhat. The villain is defeated with teamwork, deception, and by outplaying him, common to the original series. And the story raises some ethical questions regarding cheating, playing god, and marooning, again in the tradition of the show. I see the differences as more stylistic than substantial, but as I said, personal history affects my perspective.
As far as general movie principles go, music can be a strong influence on audiences, and Wrath of Khan has a great score.