There's been an awful lot of noise made about Radiohead over the last few weeks. They have now ascended to such giddy heights that all they need to do to persuade news hungry media that the album format is dead is to comment, in passing, that the process of recording a full length is tiring. They command enough serious critical respect from BBC Radio 4's Today programme to have them break the exclusive story of new song 'Harry Patch'. Their status within the worldwide music press is such that a second new track, 'These Are My Twisted Words' – initially leaked by the band, or by someone else, no one seems to know – provokes conspiracy theories that threaten to break the internet.

So how did this happen? How did these five quietly spoken, publicity-shy, private school-educated men reach the rock firmament? They consistently turn in “difficult” (though far from inaccessible) albums; they increasingly turn their noses up at catering to radio formats and shy away from the press; they work ostensibly without a record label; and they're of so little interest to the paparazzi and tabloids that you'd be hard pushed to find so much as a picture of their partners. And yet this week they headline the Reading and Leeds festivals, and tonight they're playing to 18,000 devoted fans in a specially constructed arena in parkland to the north of Prague.

The answer lies precisely in the one thing that few people have bothered to discuss for some time, arguably since the band's decision to make In Rainbows available on that 'pay-what-you-want' basis. The answer lies in their music. The fans talk about it all the time, of course. That's the reason they're here. They roar and whoop and scream before a song's first chord has rung out. They're touched by Radiohead's music precisely because it's almost the only thing that seems to matter to the band in an age where gimmick is everything. The band's haters, however, attack them for snivelling over-sincerity, for the fact that they have adopted an anti-corporate stance on the back of five major label-funded albums, for claiming to be innovative by selling their music online when others have done this for years, for suggesting that their music is challenging when compared to, perhaps, Sunn O))), they only offer bubblegum pop.

But Radiohead don't preach. They merely carry out their business of making music with the minimum of fuss, quietly drawing attention to causes they believe in and turning their back on (or at least trying to squeeze by) a system that threatened to suck the passion they had out of them once and for all. It's the media that makes the noise, you see, not the band. The band just makes the music, and it's the music that makes them. This is how they should be judged. You think they're a whiney bunch of pretentious pretenders? Fine. Move along. They make few claims for themselves. Don't waste your breath trying to justify this on any other grounds than that you don't like their records. They're not asking you to like anything else.

Tonight in Prague there are so many people who like Radiohead simply for their music that this show resembles nothing so much as a giant celebration, something that further gives the lie to the idea that Radiohead are the epitome of complaint rock. It's the first time they have performed in the country, and tickets to the front of the arena are so expensive (approaching one tenth of the average Prague monthly salary) that it is almost – and I speak from experience – cheaper to collect your car from the local pound after it's been towed. But this ensures that all present are determined to enjoy the experience despite half hour queues to the bars and the distinct feeling that getting out of this fenced-in space in the event of an emergency could be catastrophic.

They arrive on stage as the sun dips beneath the horizon, kicking off with '15 Step'. That the reception is rapturous despite the song's awkward 5/4 time signature – its beats are like hiccups, in that you know they're coming but you're never quite sure when – is a testament to the fact that it's possible to trade in something other than cookie cutter rock and still reach the masses. For the best part of two hours they continue to mine gems from across their catalogue and despite next to no communication with the audience ensure those previously rammed bars remain empty.

It's not all perfect: 'There There' appears initially somewhat dirge-like, though Yorke's falsetto breathes life into the clay; 'Weird Fishes' canters off at an unexpectedly fast pace and threatens to maul the tender, strangely Blue Nile-ish soul at its heart; Jonny Greenwood's solo in 'Lucky' seems to reveal a love for David Gilmour's more flatulent moments. But he redeems himself in 'Bangers & Mash' (, riffing at its end like a deeply paranoid Keith Richards, and even his brother Colin Greenwood's very occasional muso tendencies are easily overshadowed by his increasingly endearing tendency to bounce up and down like a baby in a romper suit. Thom Yorke, meanwhile, now dances like a man who hates to do so, has been told he has to, and has finally found out that it's fun.

But what about when it's perfect? Because, frankly, most of the time it is. Radiohead may appear morose to some for their refusal to engage in rock music's standard routines – this includes their stage show, refreshingly free of stadium rock excess and beautifully if simply lit – but their music has an ability to inspire passion of each and every kind. People dance frantically to the radio static of 'Idioteque' and the hallucinatory loops of 'Everything In Its Right Place', they lock lips to a swooning '(Nice Dream)' and an especially eerie take on 'Nude'. They rock out to an explosive '2+2=5' and 'The National Anthem''s relentless but driving bass, they lose themselves in the communal delight of 'The Bends' and the glacial beauty of 'Pyramid Song', then they welcome the shoegazing of 'These Are My Twisted Words' like an old friend.

The band's musical reach has extended itself almost chaotically from its indie-rock and grunge roots to accidental lighters-in-the-air stadium rock, from the subsequent prog tendencies on to glitchy electronica and back towards the more measured sounds of their most recent recordings. But the overall impression tonight is simply of a modest but great band at their most confidently creative peak rather than dilettantes showing off their record collections.

If a band's value is measured in terms of both their reach and their artistic qualities then Radiohead must surely be considered one of the most important of their generation. These five quietly spoken, publicity-shy, private school educated men have devoted the last two decades to crafting beautiful, elusive, chimeric music while compromising few of their principles and without sacrificing their dignity. Their success is quite simply a victory for common sense and it should be thought of as such: it's all about the music, man, and for that they need make no apologies.