Reading a little more of the text, the author seems to be building a thesis that some scientific breakthroughs are politically motivated; that Darwin and Galileo in particular were as much attempting to destabilize the establishment of organized religion as they were trying to discover new truths. The author describes this as a "fashion of thought and [...] in science."
In the sentence you ask about, the subject is "the Enlightenment belief," which refers to the Age of Enlightenment, the period in which the scientific method was established. The point of the scientific method is to establish a set of criteria by which a hypothesis or general knowledge may be tested. As the author points out, this merely tests that something is not false, not that it is true; this is called falsifiability, and without it, formal logic (the collection of rules by which we humans can empirically verify our thoughts and knowledge; in other words, how we know what we really know) simply does not work.
But falsifiability vs. verifiability is a red herring in this context. I think this paragraph is an example of appealing to motive, a logical fallacy that concludes something must be false based on its context rather than content.
Enlightenment could be seen as a loaded description, as though one were saying that science is triumphing over religion, faith, art, etc. Objectively, it's simply referring to the fact that a really huge number of things were discovered and tested during this time, largely due to the formalization and growing acceptance of science as an engine of knowledge. The book is a comparison of art and science, and the author seems to making the point that scientific endeavor can be subjective. In this context, the phrases you ask about seem to have a fairly standard meaning:
1 a : a belief or practice resulting from ignorance, fear of the unknown, trust in magic or chance, or a false conception of causation
b : an irrational abject attitude of mind toward the supernatural, nature, or God resulting from superstition
2 : a notion maintained despite evidence to the contrary
1 : the act of subverting : the state of being subverted; especially : a systematic attempt to overthrow or undermine a government or political system by persons working secretly from within
2 obsolete : a cause of overthrow or destruction
Source: Merriam-Webster, online
Political subversion would mean an overthrow of or opposition to an established regime, specifically the highly religious monarchies of 2nd millennium Europe. And freedom from would mean an absence of; that Enlightenment and science are pure and untainted by baser human motivations.
So as Mitch points out, there's nothing really surprising about the meaning of those words in this context. Going any further would be an exercise in literary criticism, far divorced from the topic of this site.