He's obviously not sick on the show, though, so what's the deal there? Well, the theory says that Caillou has stuffed animals and pets that remember him as he was before he died.
This explains why the stuffed animals are in live-action, while Caillou's stories are animated: When the show is animated, that's the stuffed animals' imagination. That explains why the show's animated episodes have that strange, white, dream-like frame around it.
So there you have it. Something to calm your soul the next time your kids are watching Caillou and you just wish it would stop.