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'm glad somebody else has brought up that TWoK sort of feels at odds with Trek despite being the best of the films. However, I say it is still the best Trek movie because:
It focuses on the characters who know from Star Trek, and their growth, change, loss, and acceptance is critical to the story. While the things that happen to them aren't limited to Trek characters, Kirk, Spock, et al. were the definition of "Star Trek" at the time.
The militaristic aspects aren't totally foreign to Star Trek. While exploration was always at the forefront of their mission, Starfleet was (as Carol Marcus pointed out) still a military organization. What has happened is that the exploration/scientific aspects in the story have been initially shifted to Dr. Marcus.
The sci-fi aspects and story telling are still very strong, it's just that Kirk and Khan shooting each other in a nebula is so great that we forget they're there.
The increased breathing room of a full motion picture that doesn't have to delve into the backgrounds of the characters we already know gives the story room to breath, and unlike TOS we have time to let events that aren't driven strictly by the "gimmick" of the scifi aspect intermingle and impact with the plot device(s).
I still hold that TMP is the most "Trek" of the movies, but TWoK is the best of the movies while still being sufficiently "Trek."