The Velvet Underground are an odd one because they do very much come from a scene, but it’s not a scene based around music like the Beatles’ scene in Liverpool or the psychedelic scene in San Francisco. Instead, the Velvet Underground’s scene is fundamentally an art scene - Andy Warhol’s Factory. Andy Warhol by the mid-60s was famous for his pop art, which attempted to find art in the products of mass market capitalism and pop culture; you probably know the famous 1962 Campbell’s Soup Cans, which depicted cans of soup, or the 1962 Marilyn Diptych featuring duplications of an image of Marilyn Monroe.

Warhol attracted around himself a scene based around The Factory, his studio, which included various bohemians, street people, transgender individuals, actors, artists and musicians - namely, the Velvet Underground. In 1966-1967, Warhol put on art events that were multimedia events - ‘the Exploding Plastic Inevitable’ - and he needed a suitable band to soundtrack the multimedia experience; this turned out to be the Velvet Underground.

Lou Reed (the Velvet Underground’s singer-songwriter-guitarist) in the earlier 1960s had worked as a songwriter/musician on staff at Pickwick Records, pumping out novelty songs and ripoffs on budget records designed to be purchased by stingy parents. John Cale, the group’s viola player, was Welsh, and had gravitated to a New York avant-garde Western art music scene that was fascinated by repetition (e.g., Cale's collaboration with minimalist composer Terry Riley, Church Of Anthrax); Cale saw commonalities between the Western art music and what Lou Reed was doing on novelty songs like ‘The Ostrich’.

An embryonic version of The Velvet Underground was formed, famously originally featuring Angus MacLise on drums; MacLise quit upon discovering they were to play a paid professional gig. Initial demos of The Velvet Underground (first released on the 1995 box set Peel Softly And See) show an acoustic group with some obvious influence from Bob Dylan and the Greenwich Village folk scene of New York, especially on the song ‘Prominent Men’. However, the Velvet Underground quickly went electric (as Dylan and some of the folk scene did) and soon found their way into being associated with Warhol and The Factory.

The Velvet Underground of 1966-1967 were very much influenced by The Factory and the people in it, and various Velvet Underground and Lou Reed solo songs are about individual members of the Factory scene (cf., ‘Candy Says’, ‘Walk On The Wild Side’). In that sense, Eno is absolutely correct with his 'scene-ius' idea - lots of what makes the Velvet Underground influential came out of their interaction with that sense. And the influence carries of Warhol designed the cover of their first album, The Velvet Underground And Nico; Nico was a German actress who had gravitated to The Factory and who became part of the band as part of Warhol’s urging. Lou Reed - especially in interviews related to his 1990 collaboration with John Cale, Songs For Drella, a tribute to Warhol, has talked about how strongly Warhol impressed his creative methods and strong work ethic into Reed; this is the theme of the Songs For Drella single 'Work').

He said I was lazy, I said I was young

He said, "How many songs did you write?"

I'd written zero, I lied and said, "Ten."

"You won't be young forever

You should have written fifteen"

It's work, the most important thing is work

However the music scene of 1967, when The Velvet Underground And Nico was released on Verve Records, was a bit nonplussed by this strange, uncompromising and often dissonant art rock with lyrics about taking heroin and engaging in BDSM; there was no real music scene outside of the Factory that the group neatly fit into. They were too loud to be folkies, too arty to be garage rock, and too much of a downer to quite fit into the psychedelic rock scene. This played a role in their comparative commercial lack of success during their career. However, eventually, in the 1970s, a scene would develop around the kind of music the Velvet Underground made - the New York punk scene - as musicians influenced by the Velvet Underground, like Tom Verlaine of Television, found each other in run-down venues like CBGBs, and as The Velvet Underground and other bands with similar aesthetics were championed by rock journalists like Lester Bangs.

Additionally, it should be said that Brian Eno himself is an enormous Velvet Underground fan and is well aware of all of this. In a recent interview with Louder Sound, he talks about how influential their sound was on him and his music:

You first heard the Velvets on John Peel’s radio show while you were at Winchester School of Art in the 60s. What impact did that have?

Within the first few moments, I thought, “Okay, this is important.” I could hear the La Monte Young influence, the sort of drone thing that John Cale was doing on the viola. I think I heard Heroin first. So I bought that album [The Velvet Underground & Nico, 1967], which not many other people did at the time. It might be hard for some people to understand, but they were a big influence on Roxy Music. Bryan [Ferry] liked them as well and we both knew about their connection with Andy Warhol, which gave them a sort of cultural position.

The mix of avant-garde musical ideas (the La Monte Young influence Eno discusses) and rock and roll was also something that very clearly came out in Roxy Music and especially in Eno's role on the first two Roxy Music records. The Velvet Underground were a big influence on a range of the more arty glam rock acts, who responded to the androgyny in the Velvet Underground (and their gender-bending themes at times) - David Bowie tried to become part of the Factory scene, and produced Reed's Transformer album; there's a song about Andy Warhol on his 1971 Hunky Dory album. Mott The Hoople covered 'Sweet Jane', etc.