There are 2 main theories about what happened at SI:
1. The PC mantled Sheogorath, became him, Jyggy went on vacation and everyone is happy.
2. The PC is bat[censored] crazy and it's all a trick.
There were attempts before this post to unify these two and say "the PC is bat[censored] crazy, but he did mantle Sheogorath". That's the path I'm taking but there's more. For most lore buffs the main problem is the ending.
When Jyggy appeared at the end and gave his speech, most people who thought they understand the isles/lore thought it was some sort of Bethesda screwup. And perhaps it is but I have a theory which makes it sensible with everything else.
Now first lets establish what actually happened:
The PC went into the isles (as a mortal obviously). He got past the gatekeeper and managed to get hired by Sheogorath himself to do bat[censored] crazy stuff. Stuff which happened to be also the road to mantling the big man himself.
When that happened, a dragon break occurred (sort of, let's not discuss the dragon's position in the isles here, all the SI books point to this) and Arden Sul (PCs all possibilities self) mantled Sheogorath and nobody could remember exactly how it happened (all possibilities). At this point Jyggylag appears and basically tells you that HE is Sheogorath the Daedric prince and that he doesn't want to fight you and blah blah.
This is the part where all the previous very logical lore of the SI crumbles to pieces. Except it doesn't.
Jyggalag was really the Prince of Order (I'm kind of omitting the Daedric word here else Proweler will pounce on my head) at the time of Mundus creation (so technically he was Daedric...). BUT! But something happened. Free will happened and he became "cursed" with it (couldn't calculate anymore) and became Sheogorath. That curse is YOU. YOU the PC and every other Arden Sul beforehand. It requires a mortal aspect IN the SPHERE to change the SPHERE and that aspect is you. That's why "you"(Sheogorath with eyes) needed YOU(PC) as the "mortal champion".
When you bathe in the fountain of madness and do all that stuff you take on the original madness of the original "curse" on yourself (and obviously become bat[censored] crazy) and then apply it in mortal form to the Isles themselves (to Jyggalag).
Now the speech makes a bit more sense. When you defeated Jyggalag's Avatar (in his own realm no less, something not possible normally) he didn't "go away". It was the Isles talking to you and accepting you as the new madness medium by which it transformed back into the good old Madgod's realm. This would also explain why your staff holds everyone with Jyggy's voice (Isles voice). It's quite possible that Sheogorath always had "A" mortal Madgod in it (changed after 1000 years when exactness tried to reassert itself).