"The dress fits."
So this isn't like "har på mig" which indicates who is wearing the dress? "Klänningen passar" could mean the dress fits me (if, or course, I wore a dress), or it could be me telling someone that the dress fits her. Also, if I meant that the dress fits me, would I need to say "Klänningen passar mig." ?
Exactly, it means that the dress is the right size. And you could well say klänningen passar mig, but it's more common to simply leave the pronoun out.
In addition, if you want to speak about the dress looking good on somebody, you'd more often use klär plus a pronoun, rather than passar - but both are correct.