I'm no spanish etymologist, but I would guess the idiom deals with "puesto" the past participle of "poner" rather than it's noun meaning. So translating directly to english, you're saying something like "I carry put a shirt." which kind of makes sense. Verb idioms always sound a little like nonsense when you try to translate them directly to another language.

The confusion here comes with puesto, which is also the participle of the verb poner; this verb is used in the context of putting on clothes, so it is this meaning rather than that of a stall that's being referenced in the expression.

Yes! You are right! The meaning is wearing something and we only use it for clothes. I'm wearing a dress => Llevo puesto un vestido

thank you all. That was very helpful. :)