I thought the Sun get's him only stronger but not get him sick too.
Luthor overcharged Superman with so much solar power that his body couldn't absorb it all, causing him to become extremely more powerful, but also have a cancer-like disease.
Well really, it's a non-canon story. That doesn't actually happen in canon.
But basically, his cells are solar batteries. They take the sunlight, and convert it into energy, giving him his power. However, in All-Star Superman, he got too close to the sun, and his cells couldn't handle that much energy, and started self-destructing.
I justify it as Lex having thrown a plan into action that we never saw prior to the All Star Superman story that screwed with Clark's ability to metabolize highly concentrated dosages of solar energy. Clark didn't know this, so he dived right to the sun to save the good guys thinking he'd just be a bit more powerful than normal for a few days. Upside: Screwed up metabolism results in more concentrated powers, i.e. SUPER SUPERMAN. Downside: Due to Clark's cells not being able to handle that level of radiation results in the aforementioned cancer.
Basically, think of it like what would happen if you ate a lot of food very quickly, and your body started trying to use it for energy before said food was digested properly. Not a perfect metaphor, but a reasonably close one.
'Course that's wild guessing with no displayed evidence beyond the fact that Superman's dying from something that hasn't killed him before. But it makes sense to me.