On Whose Authority?

Pope Francis has been speaking out against Islamic extremists, claiming that using their religion to justify violence as they do is wrong. "Authentic religious spirit is being perverted by extremist groups" he says:

All believers must be particularly vigilant so that, in living out with conviction our religious and ethical code, we may always express the mystery we intend to honor. This means that all those forms which present a distorted use of religion must be firmly refuted as false [...]

I'm glad the Pope is speaking out in this way, trying to use his influence against what is a serious problem.  But why does he think anyone should listen? The President of Venezuela is free to speak out against American foreign policy, but we're free to ignore him.  I dare say Muslims who support terror feel even less attachment to the Catholic Church.

The Pope speaks as if there's a general, agreed-upon thing called religion which has rules everyone must follow, rather than specific religions each with rules of their own--indeed, separate religions that are mutually exclusive.  The Pope is an authority on his own religion (and even there I'm sure many Catholics disagree with him), but how does he have any authority to tell other religions how to act?  If he wants to say other religions are false I can understand that, but to seemingly accept these religions and then interpret what they mean I don't quite get.