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've always maintained that TWOK is not just a really good Star Trek movie, not just a really good science fiction movie, but a really good movie period. It transcends the franchise and the genre, which I don't think can be said of any other Trek movie.
It's a movie that's about something meaningful - getting older, confronting mortality and legacy, and renewal through sacrifice. Kirk starts out feeling old, worn out, but ends with him saying "I feel young". Maybe it's a little trite when put so bluntly, but it's executed in an elegant and impactful way.
TWOK evolves Kirk in a way we hadn't seen before. He is a different person at the end of the movie than he was at the beginning (or throughout the TV show). Less cocky, more aware of the consequences of his actions, because it literally cost him his best friend.
Spock's death scene was the first time Star Trek ever made me cry. You can argue that TWOK is more militaristic than Star Trek normally was, but the themes of friendship, loyalty and sacrifice is pure Trek idealism. One could even argue that TWOK is about exploration, but an inner journey not an external one: Kirk encounters for the first time (by which I mean in a way that truly hits him) "death, the undiscovered country" (the film's working title).
In David Marcus and Saavik TWOK introduces what might have been a new generation of characters, to whom the torch might have been passed if they hadn't been killed off and sent to the home for pregnant Vulcans, respectively, in later movies. In either case, these two new characters - especially in light of Spock's death (a death sadly temporary, to the franchise's long-term detriment) - tie into the themes of mortality and legacy.
TWOK has what's arguably Shatner's finest performance - certainly in Star Trek, maybe ever. Everyone else is also in top form.
TWOK has the best antagonist. So compelling was Khan that they keep on trying to remake the "so-and-so is out for revenge" story. Montalban was so good Paramount even launched a "For Your Consideration" campaign to get him a Best Supporting Actor Oscar nod. And Montalban's Khan easily has the best chest of any Trek villain.
Even though Khan is the best villain, Khan and Kirk actually never physically meet. Their entire confrontation is carried out over comms. They never get into a fistfight or even breathe the same air, something that took me years to realise. Because theirs feels like the most visceral, tense and personal battle of any movie.
Even though TWOK was made on a very limited budget, a lot of the production design and visual/special effects hold up - there's a reason why Robert Fletcher's monster maroons are so iconic for instance. So many effective little moments. Eg the Genesis simulation (one of the first uses of CGI in a movie), though primitive by today's standards, still looks really cool because of the way it was storyboarded by ILM, with the camera sweeping ahead of the Genesis effect, then the effect catching up to it.
The battle scenes have real weight. I've always thought that Meyer's conceptualisation of starships as capital ships - rather than as jetfighters - made for better fight scenes. The entire movie is basically Roddenberry's "Horatio Hornblower in space" idea realised (hence touches like the bosun's whistle and the old-fashioned look of the uniforms), an idea which carried over to how he staged the battles, with Enterprise and Reliant squaring off like galleons at sea.
Speaking of battle sequences, the "gatling gun" phaser effect is still the best phaser effect in Trek. And you've never felt the pain of the ship getting hit as acutely as the "can opener" shot: another example of a shot which is unremarkable at a technical level today, but which still has an emotional impact. Ditto the Ceti eels.
Horner's music is arguably the best of any of the movies. There are individual tracks in other movies that might rank alongside or above the best of Horner's tracks - eg Goldsmith's First Contact theme or Giacchino's "Enterprising Young Men" - but as a whole I don't think you can surpass TWOK's score.
There are so many iconic moments and lines. "Aren't you dead?", the Dickens and Melville quotes, "I never forget a face, Mr... Chekov", "I don't like to lose", "He's so... human. / Nobody's perfect Mr Saavik", "I have been and always will be your friend", "Of all the souls I have encountered in my travels, his was the most... human".
Is it a perfect movie? No. Eg while a lot of production design works, some of it looks cheesy (like Meyer's obsession with blinkety lights). Some of the supporting characters aren't utilised as well as they might have been. (But then, the TOS movies have all been Kirk movies - it's worth remembering that TOS was not an ensemble show, but one with a clear primary, secondary and tertiary character.) But the elements in TWOK that don't quite work don't detract from its overall impact and quality.