I can walk down the street and someone will come up and say, 'Weren't you the bassist in Nirvana?'" says Krist Novoselic. "And I'll hear a story about how our music changed their life. How their parents were going through a divorce, or how they were in rehab, and the music helped them through. Or, 'I was in a kibbutz that summer, and Nevermind was the soundtrack; I had the time of my life.' The music connected with people in an incredibly personal way." He's quiet for a second. "I don't know how I can explain that."

If Novoselic and his Nirvana bandmate Dave Grohl, today leader of stadium rock giants Foo Fighters, still struggle to define the nature of their former band's unique alchemy, the world isn't sick of hearing about it quite yet. The two surviving members of the band are commemorating the 20th birthday of the group's third and final studio album In Utero. Instead of a cake, there's a box set: a four-disc, 89-track block that, in honesty, doesn't so much scrape the barrel as tunnel through its base, carve through the floor tiles and penetrate several layers of top soil. Luckily, none of this tarnishes In Utero itself, an album of caustic rock and cryptic, sometimes grotesque, lyricism that exploded like a ripe boil on the face of the mainstream. "That was Kurt Cobain's artistic vision," says Novoselic. "It was strange, fierce, kind of weird, but beautiful at the same time."

Nirvana never set out to change the world. In 1991, they were a promising punk-rock band from Washington State with a debut major-label album that might, with luck, sell in the six figures. Then MTV started rinsing Smells Like Teen Spirit, Nevermind unseated Michael Jackson's Dangerous from the top of the Billboard chart, and Kurt Cobain became the reluctant poster boy of a new sound – grunge. "The first thing we did when Nevermind went huge is cancel everything and go into hiding," recalls Grohl. "U2 and Guns N' Roses wanted us to tour with them, Lollapalooza wanted us to headline. All these offers, and we thought, 'Let's just go home and take the ball with us.' Like, game over."

The band in 1990, pre-Nevermind. Photograph: Pictorial Press Ltd / Alamy/Alamy

Nirvana's musical response was In Utero. In defiance of their label, Geffen, they called upon the production talents of Steve Albini, alternative rock firebrand behind acerbic noise groups Big Black and Rapeman. Instead of radio-friendly unit shifters, there was a song sarcastically titled Radio Friendly Unit Shifter, smothered in squalling feedback. Cobain's songs touched on fatherhood (Milk It, Scentless Apprentice) and feminism (Frances Farmer Will Have Her Revenge On Seattle, a fantasy of cosmic vengeance for a 1940s actress subjected to brutal mistreatment while incarcerated in an asylum). But it also dwelt on the gynaecological and the diseased: see the sickly-sweet Heart-Shaped Box, with its cancerous growths, carnivorous orchids and "umbilical noose". Occasionally, the album's bluntness still alarms.

It's hard to draw much holy wisdom from In Utero's tumult of anger, black humour, principle, guilt and confusion. "Nirvana were conflicted," says Novoselic. "We cut our teeth on 1980s American hardcore – intense and political music about independence from the state, independence from corporations. We were appalled by the first Iraq war, the jingoism, the petty nationalism. But at the same time we signed a record deal with Geffen, a subsidiary of this Japanese industrial electronics company. Bands like Pavement and Fugazi remained fiercely independent. We had punk-rock values, but we signed those papers. I can't sit here and give you the spiel about independence, especially knowing [Fugazi's] Ian MacKaye. I could never face them again."

From their major label vantage point, though, Nirvana reached an audience their indie peers could only dream of. "We meant a lot of different things to a lot of different people," says Grohl. "That's one of the great things about music. You can sing a song to 85,000 people and they'll sing it back for 85,000 different reasons." Ever upbeat, Grohl is optimistic about the current state of rock, thrilled to hear young bands still cite Nirvana as an influence. "We were real and visceral, fucked-up and ugly. That was what people were craving. And that will never go away. There's a band in a garage right now writing songs for an album that will do the same thing Nevermind did some 20 years ago. We don't know who and where, but it will fucking happen again. All it takes is for that storm to break."

More so than Nevermind, In Utero pointed underground – to alternative rock and the punk feminism of riot grrrl. But its influence spread outwards, too. Liam Howlett heard the gnarly riff of Very Ape, the two-minute blast that kicked off side two, and sampled it for the Prodigy's 1994 single Voodoo People. Following his death, Kurt became a lyrical namedrop for rappers from 2Pac to 50 Cent to Jay-Z, who evidently found something relatable in this nihilistic rock star and his tale of drugs, guns and untimely death. Grunge was supplanted in the marketplace by nu-metal, but Limp Bizkit's Fred Durst described Kurt as "an inspiration". And when the next Voice Of A Generation came along, you couldn't help but look at Marshall Mathers, a bleach-blond Molotov of rage, and spy something faintly familiar.

The 20th anniversary reissue of In Utero is out on 23 Sep