If ‘what’ natural theology is, is still escaping you, then let me try to explain it again; well, let me have George Hunsinger explain it for you, at least in the way that Karl Barth understood what natural theology was, and for what reason it continues to persist, especially and primarily in the Christian church. Here is Hunsinger on Barth and an explication of what natural theology entails:
[I]t is this conception of human existence as in itself hostile to grace, even and especially when confronted by God’s Word, that underlies Barth’s analysis of the persistence of natural theology in the church. Natural theology, it might be said, is presented as a theology that violates the essential precepts of objectivism, actualism, and particularism [all three identified as regulatory and organizing principles for Barth’s theological methodology]. Not mediated, not miraculous, and not unique in kind is the way our access to God appears, from the standpoint of natural theology. According to natural theology, as Barth understands it, our access to God is not something mediated exclusively in and through Jesus Christ; rather it is, at least in part, immediately open to us (and we to it). Again, it is not something miraculously enacted and bestowed (in and by Jesus Christ) by virtue of special and self-renewing event, but is rather, at least in part, at the disposal of or own innate capacities. Again, it is not something uniquely grounded in itself both ontically and noetically, but is rather, at least in part, independently and generally given to us apart from the particular history of divine self-revelation as centered in Christ. Natural theology is thus conceived as a theology according to which our access to God is not mediated but immediate, not miraculous but natural, and not unique in kind but generally given. [brackets mine] [George Hunsinger, How To Read Karl Barth: The Shape of His Theology, 99-100 nook.]
So take notice, the rejection of Barth’s alternative to natural theology, the analogy of faith/relation, for knowledge of God is based upon an anthropology that is not needy, but full of self-glory. What is built upon this exaltation of a self-glorious human existence is what, well, is ‘natural’! We would rather build our knowledge of God up on a verification process whereby we get to dictate the verification parameters, instead of allowing the object/subject under consideration to dictate those for us in ever opening and increasing ways as we engage with, well, God in Christ, in ever refreshing and more intimate ways. Natural theology is built on an human existence still clinging to the hubris that the cross left an aspect (our intellects) of our existence alive and well, and it is thereupon which we have the capacity in ourselves to provide the controls through which we engage with God in Christ. Natural theology, then, operates from a mode of fear, and the lie, wherein the human is enclosed and conquered by a posture of fear; fear that he or she might be duped by something, or someone that he or she does not have control over.
There are other things to be said, and other implications to be noticed. But hopefully you have a better feel for what a Barthian notion of ‘against natural theology’ is, because you now have a better conception of what Karl Barth was against [as what he notoriously called, ‘anti-Christ’, and you can now see why].