Today, the Grand Valley Dale Ballroom sells itself as the leading special event and banquet facility in Columbus, Ohio. "Getting married at the Grand Valley Dale Ballroom," its website says, "is Oh! So! Romantic!" At least part of the romance lies with the ballroom's history as a music venue, where Frank Sinatra and Perry Como trod the boards: "The History of Valley Dale," runs the slogan, "is a Story of Good Times."

For some reason, the Story of Good Times neglects to mention the night in 1966 when Andy Warhol's Exploding Plastic Inevitable came to town. On stage, amid strobe lights and Warhol's films, the Velvet Underground and their German singer, Nico, played a set of songs you could claim, without hyperbole, would change rock music for ever: Heroin, I'm Waiting for the Man ("it's about scoring dope," deadpanned Lou Reed), All Tomorrow's Parties, Venus in Furs. Judging by the lo-fi recording included on this six-CD box set, the audience appears to have received the lot in horrified silence.

You can mock their lack of foresight – the very sound and vocabulary of rock music is expanding before their very ears and they seem to hate it – but can you really blame them? On the most prosaic level, by the time they get to the future classics, they've already been subjected to the erroneously titled, 27-minute long Melody Laughter. It begins with formless noise, the feedback alternately shrieking and groaning like wood under stress. After about a quarter of an hour, it resolves itself into something vaguely approaching a song, albeit one decorated with improvised vocals by Nico: not, it has to be said, one of music's natural scat singers.

The giant musical screw-you that constituted their opening number aside, how could anyone in 1966 have hoped to get a grasp on the Velvet Underground's music? Some call the album on which some of these songs were released a few months later the most influential record in rock history: certainly, it's hard to imagine what rock music would sound like today if the album had never existed. But nearly half a century after it was released, it still feels mysterious and unknowable.

With hindsight, you can hear vague intimations of music that came before it. At one extreme, the drone of John Cale's viola was informed by the avant-garde minimalists with whom he'd studied before joining the band, while the piano thumping through All Tomorrow's Parties and I'm Waiting For the Man had its roots in LaMonte Young's X for Henry Flynt, which "required the performer to repeat a loud, heavy sound every one to two seconds as uniformly and as regularly as possible for a long period of time". At the other, there was R&B: There She Goes Again famously pinched its vamp from Martha and the Vandellas' Hitch Hike, while Maureen Tucker learned her ultra-primitive drumming style playing along to Bo Diddley albums. Lou Reed's obsession with free jazz pioneer Ornette Coleman must have informed some of its dissonance, while his love of doo-wop is evident in the album's gorgeous ballads: the box set contains a different take of opening track Sunday Morning, which carries a distinct hint of the eerie, reverb-drenched atmosphere of the Flamingos' I Only Have Eyes For You or Nolan Strong and the Diablos' remarkable 1954 single The Wind. But they're vastly outweighed by moments that defy any rational explication. Even today, you listen to Heroin, lurching from its measured, vaguely folky opening to screaming noise and back again, or All Tomorrow's Parties – a kind of glorious, stately din – and think: how did anybody end up sounding like this?

It's a question this box set – featuring the live set, mono and stereo versions of the album, a rough mix discovered a decade ago on an acetate bought for 75 cents in a flea market (and sold on eBay in 2006 for $25,200), Nico's solo debut, Chelsea Girl, and sundry outtakes – doesn't answer: all the alternate takes in the world can't account for the weird alchemy that existed among the band's five members. Instead, the insights it offers are small but significant. What may be the album's most bracing moment – when European Son's sneering malevolence suddenly erupts into sonic pandemonium – is revealed to be the result of a judicious tape-splice. Initially, the sound of shattering glass (actually John Cale pushing a chair into some metal plates) heralded nothing more than a surprisingly weedy guitar solo. The development of Lou Reed's vocal on Heroin, meanwhile, is fascinating. He starts out singing the whole song in a tone of bitter indifference, but by the time they reached the released version, its emotional temperature is far more complex and troubling. He coos the opening lines with a wide-eyed tenderness, like a love song; he goes on to variously sound angry, resigned, archly superior. At one point, as the song temporarily collapses into squealing chaos around him, he gives an incredulous laugh, as if he can't quite believe what he and the rest of the band are doing.

He wasn't the only one. What the box set succeeds in doing is to highlight the sheer bewilderment the Velvet Underground's innovations caused at the time. Quite aside from the Columbus crowd's stony disgust, Chelsea Girl was a game attempt to turn their dark sound and Nico's mournful lowing into something vaguely saleable, splicing it with more straightforward singer-songwriter melancholy and lavishly arranged baroque pop: for all its highlights, not least her lovely version of Jackson Browne's These Days, it's evidently the work of a record company at their wit's end. Even the Velvet Underground themselves sound a little confused at the new terrain they're mapping out – among a selection of rehearsals taped at Warhol's Factory lurks a recording of an increasingly frustrated Reed attempting to teach Venus in Furs to a baffled Nico: "It goes 'Strike dear mistress and cure his heart.'" "VOT?" That seems to have been almost everyone's initial reaction to The Velvet Underground & Nico. Now, 45 years on, it's as enigmatic and resistant to explanation as ever. All you can do is succumb to it.