Hmmmm...multi-billionaire ladies man, who is often depicted as the smartest man in the world, Sherlock Holmes and a ninja master, all while wearing arguably the best costume in superhero land with a ton of cool gadgets. No idea why he is popular. As a popular saying goes, "Men want to be him and women want to be with him"
Basically Batman is 'effin cool, he's a badass that can scare criminals just by giving a stare and I'm happy to have been a Batman since 1987, even before Tim Burton's movie I knew Batman was awesome, just check the myriad of comic book writers/artists who've all taken a crack at him proves how great of a characer he is.
This has happened before more than once.
The 1960's Batman was a phenomenon of "Bat-Mania."
In the late 1980's, people were going to the Batman movie dressed as the Joker and girls going dressed not as Harley Quinn but as a female Joker. I doubt most of them had ever read the comics but it was a cult sort of thing.
With the DK trilogy, for a lot of people, dark = cool and it's happening again.