Anyway, the girlfriend asks me why Superman's got a beard and is hitting wolves with a sword, and I'm forced to explain that Superman's powers are the result of him living under a yellow sun and they don't work if he's under a red sun. Which, trust me, sounds particularly insane when spoken aloud, but that's not the point. The point is that she then asks me if Kryptonite would still kill him were he under a red sun. I was all, "well, I mean, I guess - Kryptonians absorb the radiation Kryptonite emits more readily than human beings, so he'd probably, like, get cancer or something," a theory I'm basing entirely on the fact that Lex Luthor managed to give himself cancer by hanging onto a hunk of the stuff for too long. Also because of bourbon.
Thing of it is, I guess I always just figured Kryptonite hurt Kryptonians by being radioactive in a way that they were particularly receptive to - like, it was radioactive in general, but Kryptonians absorbed the radiation differently and more rapidly than humans in the same way that regular people under a yellow sun just get a tan - and then, upon the removal of the Kryptonite, their powers would kick in and heal them back up, so they don't end up riddled with tumors. I guess I never much questioned it - yellow makes Green Lantern rings not work, fire makes Martians weak in the knees, Kryptonite hurts Superman; you don't think about it, it just happens. Truth of the matter is, I never gave it too much thought but, true to form, Wikipedia did.
Like Hanna-Barbera's Birdman, Superman in some ways is a living solar battery; his cells absorb electromagnetic radiation from yellow stars (like Earth's sun). Kryptonite's radioactivity possibly interferes with this semi-photosynthetic process, driving the energy out of his cells in a painful fashion.
First off, only on the internet will you find somebody using freaking Birdman as a point of reference for anything, except maybe in a case like "this cartoon is slightly shittier than Birdman." Seriously, you're describing maybe the fourth-most recognizable character in worldwide popular culture, and you're going to name-check effing Birdman? Who's going to be reading this thinking "oh, he absorbs sunlight! Just like Birdman!"?
The article goes on to state that Kryptonite should have no effect on non-superpowered Kryptonians, so I guess it must have something to do with a Kryptonian's ability to process electromagnetic radiation. Still, I kind've like my explanation better than Wikipedia's, if just because I like me better than Wikipedia. You know, in general.
So, having reached no real conclusions, I reach out to you, the nerdosphere, to explain to me how the Hell Kryptonite works. Get to it.
Edited to add: I should've kept reading the Wiki - the Animated Series had pretty much the same explanation as me. Which means I clearly synthesized their thoughts as my own. Tsk tsk, me. That one still makes the most sense to me, though.