It strikes me that the in-game explanation for a limit on attunement is pretty easy-- It takes a certain amount of spiritual or mental investment to create a communion with a powerful item and that one can only make so many such commitments before losing too much of oneself or becoming overloaded.
Why 3? Well, as someone mentioned, it's as good a number as any. However, if we just consider that a baseline, I think it opens up some nice opportunities for 'modules' that expand how attunement works in the game. It would be easy to imagine that different items require different levels of commitment-- so perhaps attuning with some great artifact instantly wipes away your attunement with any other item. Or, mechanically, 'this takes up two slots!'.
One could even imagine something more granular (in a more detailed module), where you have some pool of will determined by various factors (wisdom? charisma? level? training?) which lets you commit so much of yourself to items before you begin to lose yourself. Different items would have a different cost. Maybe there's increasing negative consequences as your pool becomes tapped.
Heck, you could have this play off of other things as well-- like your ability to dominate or manipulate others. I can see it coming into play in performing a powerful ritual. Sort of an insanity point mechanic.
Obviously that's way beyond what you'd want to do in the core game. But I can see certain campaigns where I wouldn't mind having a system like that.
Anyway, the short of it is that I think it should be pretty easy to have the flavor appear to dictate the mechanic in the core rules-- part of making magic items mysterious and cool. And I think it's pretty important that the mechanics serve the flavor. Sure, it's nice that attunement gives you another way to moderate the effect of powerful magic. But when I read attunement rules I want that to feel like some side effect of this cool system for giving magic items more weight and intrigue.
AD