I love the White Stripes. I could do without the crazy brother/sister subtext, the avante-garde red/white thing, and the other pretentious trappings of the band, but the music is awesome. Jack White is a phenomenal talent. Meg is a just okay drummer, but her limited abilities fits the style of music they are playing and actually add something to it in its minimalism.
I like them for the same reason Marley23 points out - their energy. I love bands that put their heart into their music. The Who is like that. When you hear Roger Daltry scream on “Won’t Get Fooled Again”, you just know that he left his vocal cords on the studio floor when he was done. Balls-to-the-wall playing and singing. The White Stripes play like that all the time.
If you get the chance, download the video of their live cover of Dolly Parton’s “Jolene”, which they performed on Conan O’Brien’s show. They took a cheatin’ and hurtin’ country song, and turned it into an anguished blast of pain and pleading. It just gushes emotion. Or check out their live performance at last year’s Grammy Awards, when they just blew the roof off the dump.
Their cover of Son House’s “Death Letter” is better than the original, and that’s saying something. Jack White is from the Robert Johnson side of the Blues, the slightly dangerous, over-the-top, hellhounds on his tail kind of blues. Eric Clapton, much as I love him, can sound pale and inauthentic next to White. White is also a supurb slide guitar player.
Basically, for me every new White Stripes album feels a little like I think it must have felt to be a Beatles fan when a new record came out. They re-invent themselves every time, and take their music to a new place you didn’t expect. Jack White has shown he can master country, rock, blues, pop, bluegrass, and roots music. He’s a genuine musical genius, IMO. His work on Loretta Lynn’s “Van Lear Rose” gave her the best album of her career, and one of the best country albums of the last decade. The stuff he did on the Cold Mountain soundtrack was uniformly excellent. We’ll be hearing about him for decades unless he burns out or dies.
FWIW, stay away from their last album if you’re not sure you like them. It’s quite different, and there are a couple of songs on it I actively dislike. De Stijl is probably my favorite, but White Blood Cells and Elephant are of equal quality. If you like hard rocking blues, just listen to those CDs a few times until it sinks in. Then you’ll either love them or not. I don’t think there’s much middle ground.