They, like every other successful artist, were popular because lots of people bought their stuff. Some of those people bought their stuff because they were popular. It had happened before Rumours and has happened since. Personally I believe it may happen again.
Even if most of them consisted of putting Stevie Nicks in front of a box fan.
In addition to the numerous comments, here is my WAG:
there was something about the timing of the album… America had endured the Vietnam War, pulled out of Siagon a year earlier, moved on from the “free love” of the 60’s, was experiencing the growth of the Feminist Movement, and yet still loved bell-bottom pants, sex, and drama… the combination of the music, the story line, the sensuality, the accessibility, and the players made it this the perfect storm, so to speak…
They had to do something. She certainly couldn’t dance…
Oh man, I had such a crush on Stevie Nicks. Something about her look just did it for me, and that quavering voice! Swoon.
I remember a performance where she was dressed in a white outfit, which kind of reminded me of a wedding dress but…more. I almost spontaneously combusted. 
I was in my mid teens when Rumours came out, and I loved it. There is no why. I just did.
I love Stevie to death, but this is hilarious.
Repeating what others have said, but Rumours was one of the first albums I ever purchased. And I still play it. Because it doesn’t have a clunker on the entire album.
Since it was an album and not a tiny CD, it came with big pictures! And one person that stood out I those pictures was a young Stevie Nicks. I had a major crush on her, like most guys, so it made it easy to like a band fronted by a woman.
My favorit song is “Go your own way”. Clinton and the Democrats permanently ruined “Don’t stop”. Oh well.
And some were sick of both, and wanted back-to-basics rock. Which explains The Ramones. 
If you love Stevie Nicks and Lindsey Buckingham check out their pre-Fleetwood Mac 1973 “Buckingham Nicks” album: - YouTube
For me Rumours was not only a great pop album it was a great classic rock album. Songs like The Chain rock, and tracks like Gold Dust Woman just sound epic and so much different from other bands.
As others have said, great song craftmanship and a bunch of diverse talent that gelled at the right time.
I love Lindsey Buckingham, truly an underrated singer and player. My only nit with him is his guitar solos, which to me seem to consist of tuneless noodling and long stretches of single-note repetition with ridiculous guitar face-pulling.
I know - isn’t he the dreamiest? 
YMMV - I love his single-note sting-y leads. Hard to do well, and I really think he pulls them off. And yes, he is a ninth-degree high Master of Guitar Face. I think of Lindsey Buckingham as kinda like Picasso - a brilliant master, totally feeding off his own ego to fuel and support HIS vision (and pulling it off), and with a misogynistic streak to him. The Guitar Face stuff just kinda fits with that profile…no clue if that has any traction…
WordMan’s first post pretty much sums it up for me. I’ll just say that Buckingham is amazing in terms of playing, producing and arranging and the guy has raw musical talent by the truckload. Pick up one of his solo albums and you’ll hear a mad scientist experimenting in the studio with all kinds of crazy ideas that are often just on the right side of the line between genius and insanity. Plus, his guitar playing is just extraordinary.
Fleetwood and John McVie are just a really powerful rhythm section who actually contribute substantially to the band’s sound. I’ve seen both Fleetwood Mac and The Mick Fleetwood Blues Band live and Fleetwood is an absolute madman who always brings his A-game.
So, already you’ve got the trifecta for drawing in people who love great musicianship.
Nicks was/is hot as anything and had a mystique about her and her lyrics that was the stuff of urban legend. Right there is marketing gold. And yet she was strong and powerful and sensitive all at once. It’s the ultimate hook. Buckingham’s songs had tension and angst and anger; Nicks’ songs had soul and mystery and love and danger. It was a great mix.
Christine McVie added broad mainstream appeal that smoothed out the rough edges of Buckingham and Nicks. “Don’t Stop” and “Songbird” really get the MOR crowd on board after the hardcore introspection of team Buckingham/Nicks. To borrow a phrase from Derek Smalls, Christine was the luke-warm water between the fire and ice of the other two songwriters. And while that might sound derogatory, she really wrote some classic pop/rock that served as the glue to hold the whole thing together.
Finally, the band was a living soap opera. Rumours was essentially “Sex, drugs and rock ‘n’ roll: the concept album” two years before The Wall, just without hammers and an animated talking arsehole.
Keep in mind that my two favourite bands are Yes and Pink Floyd. It’s not like I spend all day listening to Air Supply. But I think that people either forget or don’t realise that Fleetwood Mac were more than just a pleasant pop/rock act.
It’s wise to go with one’s strengths.
Remember Neil Young’s guitar solo in “Cinnamon Girl” off of “Everybody Knows This Is Nowhere”?
Hey, it’s two different strings :).
For the same reason that Surrealistic Pillow, Joshua Tree, Graceland, Nebraska, Four Way Street, Deja Vu, Hotel California and any of fifty or so other great albums worked; great musicians playing great songs.
Some of the same reasons people love Soap Operas… TV Dramas… Real Housewives and all those “reality” relationship confrontation shows… People have a never ending interest in others break-ups… fights… make-ups and etc.
Fleetwood Mac did all that with great songs and good music… Christines lyric on 'Over my head"… you mood is like a circus wheel… you’re changing all the time
Men and Women… package it up… it never fails to deliver… its in the DNA… example why lenny Kravitz’s second album is ten times better than his first…
Sorry: insufficient. There are too many examples of great musicians playing great songs that go nowhere in their day and either stay nowhere or become cult classics, like Television’s Marquee Moon, Big Star and countless others.
Yes, great musicians playing great songs is a key ticket to entry, but you may not get any further than that…
Radio-friendly crossover. Soft enough to be considered soft rock, but with enough energy to crossover and get played on rock stations.
Like others have said, you have to consider the context - disco, rock that tended to the hard side, and soft rock that was treacly.
Fleetwood Mac filled an open space.
Plus, good songs, great production, two women in the band.
For those of us with Depression-raised parents, it was difficult and tricky to get a rock n roll album into the house sometimes. It was helpful if the band in question had been on the Mike Douglas show and looked not too Mansonesque. You know, like Fleetwood Mac.