The best explanation I've read came from Anthony Bunn, the editor of the Stoke City fanzine, A View To A Kiln. Back in 2001, he told The Guardian:
"Basically it was started in the late 80s by a Stokie nicknamed TJ. It's unique due to how its sung. It starts off by the crowd singing "TJ! TJ!" to get him on his feet to start us off. We then go deathly quiet whilst TJ duly starts the song on his own. The rest of the Stokies only join in at the end of each line until we reach the "she stood there laughing" bit. Then the chorus is sung twice. At the end, if it's a rousing rendition, we all clap each other (don't ask me why).Sounds strange but it is a truly awesome sight and sound when done at full tilt. That's probably why most team's fans try (and fail) to drown it out with whistling."
The song's links with Stoke City fans apparently extend back to the 1970s when a rock band, The Sensational Alex Harvey Band, played a gig supporting supergroup Yes at the club's former home, the Victoria Ground.
The Scot Alex Harvey performed quite a dark, macabre version of Delilah which reached No 7 in the UK charts in 1975. He performed it at the concert.
Stoke played the next day and a group of fans began singing it. It slowly caught on, with the lyrics being amended by fans on the terraces of the Boothen End.
More recently some parents have objected to the 'erotic twist' given to the line "I put my knife in her hand" claiming it upsets their kids.