Feet sweat, and can stain carpets and such after a while. It's the same reason one wears gloves when handling valuable things, etc.

> Why would it be unclean for others? That makes no sense.

The claim was that I was tracking in all the dirt from outside on my feet, and spreading it around where it would dirty up others. Since I'd be doing exactly the same (plus all the other dirt from years of wear) if I were wearing shoes, I agree with you that it makes no sense.