Dignitatis Humanae makes a narrower claim than it is usually credited with, and the narrowness is what allowed it to pass. It does not say that one religion is as good as another, and it says explicitly that it leaves untouched the traditional Catholic teaching on the moral duty of individuals and societies toward the true religion. What it asserts is a right held against coercion: that in religious matters no one is to be forced to act contrary to conscience, and no one is to be prevented from acting according to conscience in private or in public, within due limits.
The ground of the right is the dignity of the human person, known both by revelation and by reason. That move matters. Earlier Catholic argument had run from the rights of truth: error has no rights, therefore the state may restrain the propagation of error. The declaration relocates the question from the content of belief to the standing of the believer, and a right that belongs to the person survives the person being wrong.
It then draws the civil conclusion. This right is to be recognised in the constitutional order of society and made a civil right. The state's competence in religion is limited to public order, and it may not command or forbid religious acts. Religious bodies are to be free to govern themselves, teach, worship publicly, and take part in public life.