The wiki article on it points to early examples of certain nightclubs having swing-dancing competitions, or big-band performances appearing in films like The Mask, but these seem more like symptoms than causes to me. For a few years (it feels like it was 1997-98, in my mind, but I'm sure it lasted for longer), you could hardly go anywhere without hearing the Cherry-Poppin' Daddies or Brian Setzer or one of the other groups that crept in on this. Even the success of something like Lou Bega's "Mambo No. 5" seems like it was on the tail-end of this as an associated phenomenon.
But why then? What made the mid-90s the right time for this weird, short-lived renaissance to occur?
Follow-up: From what I remember from researching some of the key swing revival bands on Wikipedia, it seems that quite a few of them came out of the third-wave ska scene.
Why would that be? Was it simply because third-wave ska bands already used many of the brass instruments employed in big band music, thereby making the transition easy?
I can't really point you to a conclusive social analysis as to why swing inexplicably struck a chord with the American public from roughly 1996-1998, but I've done a significant amount of research on the subject (yes, the Cherry Poppin' Daddies Wiki - and its 160 references - is a product of my own online ennui), so I'd like to think I know a thing or two about the movement, and it actually has a somewhat coherent timeline of events leading up to it.
Starting with neo-swing itself, I've never heard any different that Royal Crown Revue were the true originators of the neo-swing scene, having formed in 1989 as an affectionate tribute to classic swing and jazz before ultimately becoming a staple of the LA nightlife (of which Big Bad Voodoo Daddy would later join, as would San Francisco with Lavay Smith). By the time RCR appeared in 'The Mask' in 1994, neo-swing hadn't experienced any crossover commercial success, but traditional big band and pop standards were assuredly making a mainstream radio comeback: Harry Connick Jr. was a platinum-selling star, Tony Bennett made a huge splash on 'MTV Unplugged', Sinatra released his 'Duets' albums with A-list guests, and there was even a "lounge revival" with bands like Combustible Edison and Pink Martini gaining chart success and hipster credibility on indie labels like Sub Pop.
When Kurt Cobain committed suicide in 1994, a lot of music journalists signified that as the definitive end of the grunge rock era, and both fans and modern rock radio seemingly started turning away from mopey alt-rock in favor of "lighter" fare: pop-punk was the next subsequent craze, with Green Day, The Offspring and Rancid all going platinum in '94 and '95. The near-coincident release of Rancid's "Time Bomb" and No Doubt's 'Tragic Kingdom' are usually credited as finally opening the mainstream floodgates for the thriving underground ska scene, and by 1996, horn-drenched, pop-punk influenced ska bands like Reel Big Fish, Goldfinger, Less Than Jake, No Doubt and Sublime were all over MTV and scoring hit records on major labels.
Perhaps because peppy horn-driven ska was topping the charts, mid-to-late 1996 proved to be the right window of opportunity when swing was accessible enough to "break": the Squirrel Nut Zippers struck big-but-not-huge radio gold with "Hell" (ironically, not swing but calypso) and the 1996 indie smash comedy 'Swingers' lent some valuable mainstream hype to the LA swing scene. From there on, it was simply a matter of one horn-fueled dance music replacing another. According to Cherry Poppin' Daddies singer Steve Perry, ska and swing pretty much co-existed in 1997. The Daddies themselves were primarily known at the time as a ska-punk band who played some swing songs; by late 1996, they noticed crowds were responding better to their swing music, so in early 1997 they slapped all of their swing songs onto one album to make a quick buck and, much to their surprise, it became the first swing album to break the Top 40 on the Billboard 200. I wouldn't say they're directly responsible for the swing revival, but if there's any group that bridged the gap between ska and swing, it was them.
When 1998 rolled around, swing had not only replaced ska as pop music's most unlikely force, it had pretty much demolished it from the charts. The big drive behind that, though, was that swing was definitely more commercialized than ska ever was, which is why we all remember "hearing it everywhere". Ska-punk was still rooted in punk and Gen-X snarkiness, whereas swing appeared fun and happy and wholesome - something Little Timmy AND his parents AND his grandparents could all enjoy (and consume). Thus is why we had the Louis Prima Gap commercial and Big Bad Voodoo Daddy playing the Super Bowl and The Brian Setzer Orchestra playing Christmas at Rockefeller Center and the Cherry Poppin' Daddies playing Rockin' New Year's Eve, etc. Compared to ska, which had spent decades across the globe developing and evolving as a versatile and multi-faceted genre, "swing" was pretty much a half-baked "product" led by only a handful of very different bands who were never part of any cohesive scene together. And yeah, I know that's mostly opinion and speculation, but there's really no real academic answer to "why was this fad popular?", is there?
And of course, by 1999, swing had completely evaporated from the charts. The first two months of 1999 introduced us Britney Spears and Eminem, inciting an entirely different wave of irritating music. :P
Okay, so TL;DR: musical trends tend to operate in reactive cycles. For example, the pompous complexity of progressive rock was succeeded by the raw, stripped-down approach of punk. Just as grunge was the voice of Gen-X apathy and disaffection, I feel swing was the ultimate rebellion against that, embracing baby boomer culture rather than deconstructing it. That's one way of looking at it, if I wanna sound smart...on the other hand, especially nowadays, swing is just a fun pastime for people who enjoy the music, style and dancing of a bygone culture, no different from the rockabilly or Tiki subcultures, of which there's considerable crossover. Don't think it gets any deeper than that. It was just something entirely new and different, albeit something that was riding on the coattails of its previous, similar-ish fad.