At an open-air concert on Hampstead Heath I was dancing to Björn Again, the Abba tribute band, when I noticed something very peculiar. It wasn't just that my children seemed oddly unembarrassed that I was belting out the old songs like a teenager. It wasn't just that my old friend Michael was wearing a pair of pink fluffy antennae with flashing lights on the end. It wasn't even the sight of a senior Times executive in the front row swaying, as if in a trance, and silently weeping to Fernando.

No, what struck me as odd was that my daughter, aged 8, seemed to know every word of every Abba song. How was this possible? I had never known her listen to the music. We do not play Abba in the house except at dinner parties that have got out of hand. Yet there she was, chirping along to everything - even the fiddly French bits of Voulez-Vous.

This, it struck, me is the true genius of Abba: the words. “I've often wondered, how did it all start? Who found out that nothing can capture a heart like a melody can?” Sure, the brilliantly crafted commercial pop tunes have endured, in a way that the spangled pantsuits and beards, thankfully, have not. But the lyrics are the key to Abba's immortality.

Listen to an Abba song once, and the chorus stays with you. Hear it twice, and you can recite it word-perfectly. But an Abba song heard three or more times remains permanently imprinted on the brain: what the Germans call an Ohrwurm, an ear worm that wriggles out again whenever the first strains of Dancing Queen are heard.

Benny and Björn were not Lennon and McCartney, let alone Bob Dylan. But the very simplicity of the words is part of their magic: they describe everyday feelings, in plain, one-dimensional words. Abba mastered pop language perfectly, in large part because English was not their first language.

No one ever struggled to understand the meaning of Abba (except when Agnetha and Anni-Frid could not actually pronounce the words they were supposed to be singing), but one can spot unexpected profundities if one stares hard enough.

Waterloo, for example, won the Eurovision Song Contest in 1974, even though it refers to a military victory that only the British care about. “My my, at Waterloo Napoleon did surrender.”

One Abbaologist has suggested, tongue-in-cheek, that this is “clearly an attempt to recontextualise 19th-century European geopolitics. Napoleon had so subverted the principles of the French Revolution that for most Frenchmen his defeat was the only way civilisation could be saved.” I feel like I win when I lose. How typically French.

“The history book on the shelf/ Is always repeating itself.” This, surely, is an echo of the philosopher-poet George Santayana: “Those who cannot remember history are condemned to repeat it.” (Actually, those who cannot remember history are frequently condemned to repeat George Santayana's quotation.)

We need not look so deep. Waterloo, is, of course, a simple love song, about giving in to the demands of love, with the paradox that this involves a sort of defeat by the person surrendering.

Much of Abba, for all the jaunty tunes and exuberant beat, is distinctly bleak. These are, after all, Swedes. In Knowing Me, Knowing You, two estranged lovers look over the wreckage of their romance: “Walking through an empty house, tears in my eyes/ Here is where the story ends, this is goodbye.” Winner Takes it All is about the pain of separation: “I figured it made sense, building me a fence,/ Building me a home, thinking I'd be strong there./ But I was a fool, playing by the rules...”

There is a vein of feminist resilience amid all the dancing and jiving and having the time of their lives. The women suffer when love fails (“How can I even try to go on?”), but then pick themselves up again. Money Money Money is a furious attack on financial greed, with the ultimate gold-digger's lament: “And if he happened to be free, I bet he wouldn't fancy me.”

Part of Abba's inspiration was to imply a romantic backstory. It is probably safe to assume that not one in a hundred people listening to Fernando knows that it refers to the Mexican Revolution of 1910. And it does not matter: all the listener remembers is that there is something in the air that night.

Abba crafted their songs with extreme care, and they knew what they had achieved. When the show Mamma Mia! was first mooted, Björn Ulvaeus made one rule: “You cannot change the lyrics... the story is more important than the song.” While other pop stars were simply shoobedoobedoing, Abba were telling stories, in a pop-music lingua franca.

“A lot of the songs were little stories within themselves,” said Björn. “They're good to illustrate different situations in life.” Which is why, unique among pop bands, their words can be reassembled into an entire stage show, and now a film.

It doesn't do to overintellectualise Abba. Some of what they sang was pure gibberish. Sometimes their grip of English simply collapsed: “Now we're old and grey, Fernando/ Since many years I haven't seen a rifle in your hand.”

Many of Abba's pop contemporaries died immediately after flowering: Buck's Fizz went flat almost overnight, the Davids Essex and Soul are just small, painful memories. But Abba are impervious to age, and not just because of the warm fug of 1970s nostalgia they impart. Our children sing and dance to Abba, because four unlikely Swedes with little personality and too much facial hair, hit on a recipe: intimate, catchy little songs with stories that were universal, and words that were unforgettable, often infuriatingly so.

Thank you for the music: but thank you, even more, for the lyrics.

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