It’s important to remember the context. MASH was based on a movie (itself inspired by a book), which was only nominally about Korea. In actuality, it was intended as a statement about the present war in Vietnam. The television series debuted in 1972. So, not only do you have typical sitcom or television fandom at work as viewers grow yo love certain performers and characters, but it was a series that dealt with the societal unrest and feelings about America’s involvement in a foreign war. The series lasted through the decade as the nation adjusted to vets coming home (or not coming home), a distrust in our government, and confusion about how best to help people. MASH was a perfect combination of sitcom and medical drama with that emergency element as well as social commentary. At the same time you had popular sitcoms like All in the Family and dramas like Emergency!

MASH was also unique in that later in its run it wasn’t afraid to be more dramatic and even had episodes without laugh tracks, which was unusual at the time. The series finale was a 2 hour special, and it is this single episode which is the highest rated. Again, this was not about the akorean War, which lasted only 3 years. MASH ran 11 seasons, spending over 12 years at war (one episode spans a year’s time). While it used Korea as a backdrop, it was really about our nation struggling to cope with our recent decade of foreign warfare in Vietnam.

On top of that, of course, there were only three networks and cable was brand new, so television ratings were higher anyway. But combine the typical high ratings you would have for a popular long-running series ending with a full-length movie, essentially and that boosts ratings. But this was 1983, post Nixon, post Carter, as America starts to deal with life after Vietnam (with movies like Coming Home and The Deer Hunter being released a few years earlier) and saying goodbye to MASH in an episode about saying goodbye to the war was a real cultural moment; a sort of collective catharsis.

EDIT: I’m typing this from bed so I know there are sources ai don’t have in front of me, but will add the few links I posted in another conment:

Here’s a 1980 BBC radio special discussing the popularity of the series with some if its creators that hits on some of this stuff.

Here’s a vintage New York Times article about the ratings for the finale. It discusses how the length of the finale contributed, since it was a two and a half hour slot, and the ratingsare based on viewership of any part of it.

And here’s a Daily News retrospective on the series from 1983 when the finale aired.

EDIT TWO: regarding the movie MASH and it’s establishment of the movie as a Vietnam statement, check out the audio commentary on the DVD. There are also bonus featurettes on the old Five Star Collection edition which go into it. I can’t cite direct quotes because unfortunately my copy is buried in a box somewhere, but they definitely go into it.

EDIT THREE: Also, just a comment for OP who mentioned growing up on Friends, even Ross on that show in one episode mentions his videotape that was “half the last episode of MASH and half the hostages coming home.” Interestingly, his ex-wife taped over the MASH episode. Doubt there’s significance to that, but found it funny that even characters on OP’s shows were fans!

MASH the TV show was the culmination of a very convoluted process of evolution. First you have MASH the book, published in 1968. A quasi-autobiographical novel co-written by an actual surgical doctor (Dr. H. Richard Hornberger) who worked in a MASH unit during the Korean War. The novel was by turns irreverent and serious, and many of the core elements of MASH as it has come to be known were there, especially the main character of Hawkeye, a jazzed up fictonalized version of Hornberger himself. If you needed a frame to understand the novel, something like a typical college romp interspersed with more serious overtones about death and war would be a good place to start.

The novel was mildly (but not wildly) popular and was optioned for a film, as almost all books of any degree of success are. Many directors passed on the film (over a dozen) until it landed in the lap of Robert Altman, who at the time had mostly done TV work. The film acquired a modest budget ($3 million, about $22 million in today's money) and a decent stable of reasonably established actors. The film was a mess, both in shooting and in execution, and it took some liberties with the source material. But what it was more than anything else was a deeply subversive (and thoughtful) anti-war and anti-establishment take, especially against the Vietnam war. The movie came out in 1970 near the height of the controversy over the Vietnam war and it's anti-war/anti-establishment message resonated incredibly strongly with the public. It was a box office smash, earning $82 million at the box office (half a billion in 2018 dollars). Some people liked the comedy, some liked the acting, some liked the setting and themes, some liked the subversiveness, it had almost something for everyone. It won the Palme d'Or at the Cannes Film Festival and was nominated for five oscars (though it won only for Best Adapted Screenplay).

On the heels of success at the box office a TV series was planned (starting in 1972). The show was significantly less openly radical and subversive than the film, but it was far from conventional, all those elements were still there, and still strong, just slightly moderated. The result was that the show was even more of a "something for everyone" product than the film. It had war, it had gritty action, it had comedy, it had drama, and it still had that anti-war and anti-establishment subversiveness streak running through it. In a time when there were only 3 major broadcast television networks, and your options for television watching were limited, MASH proved to be popular with an enormous section of the viewing public.

The show's popularity fed on itself. As the show ran on it got more polished, more sophisticated, and more inventive. Some of the episodes are hallmarks of transformations of what sitcoms were capable of at the time, pioneering new ways of storytelling in the television format. It wasn't just the same gags and jokes, or even the same drama formulas, throughout the years, the show kept topping itself again and again, sustaining its popularity over a run that not only outlasted the duration of the original Korean War the show was ostensibly about (which lasted only four years) but also outlasting the end of the Vietnam War in real-time (America pulled out in 1973, Saigon fell to the North Vietnamese in 1975, and MASH continued on the air until 1983). By the end MASH wasn't just a popular TV show, it was an institution. It had spanned four presidencies and all the tumult of the transition from the 1960s to the '80s, it had seen a war end, it had seen a presidency fall from corruption and abuse of power, it had seen the world change. And along the way it had rewritten what it meant to be a sitcom, what was possible on television, racking up awards year after year. Music had The Beatles, television had MASH.