Putney Bridge, south London, December 2010. Thick snow clings to the banks of the Thames and icicles hang from wrought-iron railings. The xx used to call this their manor, but they have been away touring the world for the past 12 months - a year whose peak came in September, when their debut album, xx, beat Paul Weller, Dizzee Rascal and others to win the Barclaycard Mercury Prize. The band's melancholic songs, with lyrics about loneliness, lust and love, have struck a chord in Britain in particular, providing the perfect soundtrack to our troubled times. The BBC chose their song "Intro" for its election coverage - and their popularity was even latched on to by the Tories, who used their music without permission at the Conservative party conference in October.

Tonight, The xx are home to do something deeply un-rock'n'roll: turn on the local Christmas lights. "They probably wanted Barbara Windsor, didn't they?" laughs Oliver Sim, the tall, handsome singer and bassist, who shaved off his old, Fifties-style quiff when the band finished touring in October.

Next to him is the singer and guitarist Romy Madley Croft, Sim's best friend since they were toddlers. She smiles shyly under her severe, triangular fringe and explains that they are here because they owe it to Putney. "Especially given all the time we spent lurking around Greggs," she laughs. The third member, Jamie Smith, who met the others in the playground when they were 11, is now one of the UK's most respected young producers and DJs. He is a quiet soul, who most often nods mutely in agreement at what his bandmates have to say.

What is striking about these three, whom I first met 18 months ago, is how humble they have remained throughout their rapid ascent. Even now, where others in their position would happily put themselves forward as spokespeople for this or that issue, The xx are characteristically reticent. This is the third time I have interviewed them, yet even now it feels like we're playing mind games. When the Dictaphone is off, they are warm and talkative; as soon as it is on, they become cautious.

Never is this more so than when we move on to the subject of politics. When The xx's music was used at the Tory conference, a represen­tative of their record label, Young Turks, posted an angry message on the social networking website Twitter. "The xx . . . didn't approve the use of their music at the party [conference] and certainly don't approve of said party," he wrote at the time.

Now, however, they quickly clam up when asked about it: Sim will only express his irritation at their name being used to confer kudos on anyone at all. When I ask if they voted, they stare at each other nervously and grimace. After a long silence, Sim tries to explain why they do this. "It's protection, I suppose," he says. "Not to flatter myself, but I know that you can influence people's opinion as a musician and I don't want to." Madley Croft nods. "Every one of us has our opinions but we are quite private people. We wouldn't want to be associated with anything."

Later, they tell me that they did vote and that they care deeply about the future of Britain - particularly its youth. But The xx desperately want to avoid being held up as moral icons or ambassadors for a generation. As Madley Croft reminds me, they made an album full of personal sentiments that they initially thought only "four people would hear".

Putney is where I first met The xx. This was pre-fame, when they used to rehearse in a tiny space under the railway arches, where sticky carpet covered the walls and empty Coke cans littered the floor. Baria Qureshi, then the fourth member of the band - whose departure last autumn Madley Croft still describes as feeling "like a divorce" - was there, too. They were all dressed head-to-toe in black, as they still do. They talked about the different artists and genres that they loved - Chris Isaak and Mariah Carey, hip-hop and dubstep - as trains rattled the walls. "That room was like a womb," Sim says now. "I miss it."

Now The xx move in a very different world. Their album recently went platinum (325,000 copies have been sold to date) and its songs have been covered by stars as various as Damon Albarn, Shakira and Orchestral Manoeuvres in the Dark. That they have achieved this success while keeping themselves to themselves is quite a feat in our flaunt-yourself pop culture. Instead, they have worked their way into our collective consciousness by inventing an urban sound that unites the emotions of indie rock with the drama of dance music.

Madley Croft is astonished at the acclaim they have received. "I'm baffled. After all this time, I just think: 'How?' We made this record for ourselves - to try to make music that all of us would like." Perhaps their success is due to the way their combined influences tick a range of critical boxes - or, less cynically, maybe the mood of intimacy and familiarity they inspire is something that people crave, especially in a world bombarded with 24-hour content.

With the music industry dominated by sons and daughters of rock's aristocracy and by privately educated musicians (according to a recent survey by The Word magazine, 60 per cent of acts that charted during one week in October had been to public school), it is refreshing that The xx come from very ordinary backgrounds. Brought up in lower-middle-class families - Sim in a council house at Clapham Junction - they went to the Elliott School, a comprehensive in Putney Heath that also counts the electropop band Hot Chip and the dubstep artist Burial among its alumni.

It isn't the state equivalent of the Brit School, however, says Madley Croft, who thinks her school's influence has been overplayed in the media. "A teacher from Elliott who had never even taught us said how great we were. It's a bit annoying. We were left alone, more than anything - although I'm sure that helped us in its own way."

After leaving school, they signed to a small indie label, Young Turks, which had the financial backing of XL Recordings, home to Radiohead and the White Stripes. Unusually for a new group starting out in the modern record industry, the band was given time and space to develop. They decided on the name "The xx" because it suggested so many things - chromosomes, kisses, pornography and even their ages - each pushing 20 - at the time of
the album's release.

By day, they worked in chain stores such as Costa and Uniqlo, making their album at night. Sim and Madley Croft would record their vocals straight on to their laptops at home, trying not to wake their parents; Smith mixed the songs in a tiny room under the XL offices in Notting Hill. Its dark, moody sound deepened as a result of this nocturnal process. It is no coincidence that xx ends with songs called "Night Time" and "Stars".

Although the album received good reviews, the bigger boys in the media didn't get The xx straight away. Radio 1 didn't seem very interested in playlisting their singles and the NME didn't think that they were cover material. Even after I accompanied the band in November last year to New York, where I watched fans queue outside for returned tickets and Courtney Love fight her way backstage to meet them, I was told by an NME editor that they weren't quite "right" for heavy coverage. The xx were not conventionally glamorous, after all, nor were they interested in pompously touting themselves as the next big thing.

But by March 2010, strengthened by constant touring and some astonishing live gigs that were lit by the band's simple, black-and-white, x-shaped light boxes, the band sold out two nights at the Shepherd's Bush Empire in London. That month, The xx finally got their first NME cover, which blazed, without irony, that they were "the most underrated band in Brit­ain". What's more, their music was pricking the ears of television music programmers, whose choices of soundtrack material are now highly influential in the music business. The opening track from xx, "Intro", was picked up for a BBC general election ident that ran throughout the spring. Its menacing notes complemented the BBC's smoke-clouded images of British hospitals, schools and soldiers perfectly. In May, the band was invited to perform live at the close of BBC2's Newsnight Election Special.

Looking back, Sim is amused by the brief period in which "Intro" became "the Rocky theme tune of politics". He was impressed by Jeremy Paxman, too. "He's quite a smooth guy," Sim laughs, noting that the presenter even played Smith's electronic drum pads just before the band's soundcheck. But they did the show largely out of curiosity, he adds. "You're asked lots of times to be on Jools Holland but only so many to be on Newsnight."

Then, after a triumphant performance at the Glastonbury Festival, came the Mercury Prize, the shortlist for which was announced in July. Due to a policy of avoiding reading their press - Sim tells me that the most he will do is occasionally flick through the scrapbook of cuttings his father keeps - they were unprepared for all the media attention, particularly after they were named the bookies' favourites.

As a member of the Mercury judging panel, I had spent the summer listening again and again to xx. Much as I tried to fight temptation, the other shortlisted albums often sat forlornly on my speakers. Each time I listened to it, I found something new: the gorgeous, building drones in "Fantasy"; the vast silence and space in "Crystalised"; and how the rough edges of Sim's voice rubbed against the sweetness of Madley Croft's.

At the award ceremony, which took place on 7 September at the Grosvenor House Hotel in London, I was worried about how the band would react when they won. I was relieved to see them accept their prize with warmth. My lasting memory of that night is seeing the trio head off into the night with their arms tightly around each other, and yet, true to form, they admit that they would have preferred to have been somewhere more modest.

“We'd hired a bar nearby for our friends and family," says Sim, "where there was a big video screen which played the announcement live. Our soundman recorded what happened and we watched it later - my mum there, running past the camera, crying, as if England had won the World Cup."

Through the summer, and The xx's post-Mercury autumn, the band's confidence appeared to grow: in live performances, Madley Croft would sing more boldly, Sim developed an onstage swagger and Smith began experimenting with different rhythms and textures. And Madley Croft revealed, to a small online magazine called Tourist, that she was gay. "I outed myself to the whole world on my friend's tiny little blog," she says shyly. "I forgot that everyone could see it. I'm proud of having a girlfriend; it just doesn't have anything to do with my music." Sim, who was also described as gay in the band's early interviews, has said nothing since about his own sexuality, and that is unlikely to change.

For me, this guardedness adds depth to The xx's confessional songs, suggesting that extra­ordinary emotions lurk behind such ordinary lives. Perhaps that is what their fans most respond to: how they articulate the drama and power of that inner life.

Back at the Christmas market in Putney, it's nearly 6pm. Hundreds of local people are gathered to watch them flick the switch, including Sim's father and old friends who find the whole business hilarious. The three friends walk on to the stage as their host counts down to zero and then turn the Christmas tree behind them all silver and gold. Sim politely thanks Putney for having them, wishes everyone a merry Christmas and poses for pictures with Madley Croft and their fans. Smith hides behind a marquee, looking sheepish.

Tonight is The xx's last press engagement for the near future and, as they leave, they engulf me in bear hugs. They must be relieved to be stepping off the promotional treadmill, but their warmth feels sincere all the same. They are not a band about pop's flash and burn and they never were. They understand, you can tell, that there is no place like home. l