This synopsis will contain spoilers!
Rather than attempting to summarize the "proofs" that Lewis offers for each of the points he raises, I will just try to summarize the actual points he is trying to make. He is too efficient with his words and arguments to attempt to summarize them adequately. One will be served better by reading the relevant portions of the text.
First, Lewis defines what Epic Poetry is, and specifically the type the Milton was writing, how it evolved from the previous types, and why Milton decided first to write and epic poem, and then chose the subject afterwards. A key element here is the idea that Milton must use language to set the scene an actual epic poem recital would have historically done.
Next, he deals with the claim that only the greatest poets of an age should be allowed to critique Paradise Lost by pointing out that if this were the case (that only great poets can identify great poets), then how would it be possible to ever know who the great poets are?
Lewis next details what it means to be a primary epic (such as what is spoken about by Homer) in which an epic was recited to a king in court (either comic or tragic). This is where more is discussed related to the need for Milton to use language in such a way to elicit the gravity and sincerity of the moment the way the court proceedings would in the time of primary epic. This is because Paradise Lost is a secondary epic, and thus only has language at its disposal.
Lewis points out while discussing epics that one must not look for single good lines, but must allow the work to continue moving forward. It is not like a normal poem in that you hunker down and analyze all the smallest details. "It is not built up of isolated effects; the poetry is in the paragraph, or the whole episode. To look for single, 'good' lines is like looking for single 'good' stones in a cathedral" (21).
As he continues, Lewis discusses the specifics of Milton's style as a secondary epic, including what he must do to evoke the necessary setting for his work (chapter VII). Lewis then proceeds to defend this style, in particular by pointing out how important it is to remember what it was like when Milton wrote Paradise Lost. In particular, at that time there were certain "Stock responses" to elements that Milton would have expected every reader to have. As example, Lewis lists "love is sweet, death bitter, virtue lovely, and children or gardens delightful" (57). He also defends the vaulted style of writing by claiming that Milton did this intentionally in order to let the reader know "that something out of the ordinary is being done" (59).
Lewis this criticizes the Doctrine of the Unchanging Human Heart (humans are ultimately the same throughout the ages) and claims it has no place in a reading of Milton (IX). Next, he discusses the theological similarities between Paradise Lost and St. Augustine (X), and points out that "The Fall is simply and solely Disobedience - doing what you have been told not to do: and it results from Pride - from being too big for your boots, forgetting your place, thinking that you are God" (71).
In XI, Lewis emphasizes how important the concept of Hierarchy is in Paradise Lost, in the sense that everything has something it is designed to serve, and that which it must rule over. Satan, of course, being an example of one who attempts to rule over that which it is subordinate to (God).
XII discusses the aspects of theology in Paradise Lost, noting those ideas which are not heretical and present in the work, heretical and not in Paradise Lost (but perhaps in other works by Milton), heretical but not in anything by Milton, and finally heretical and in Paradise Lost.
In XIII and XIV, Lewis details the character of Satan (and why he was not someone Milton would have considered a hero or someone to take pleasure in) and Satan's followers. XV discusses some mistakes that modern readers make about Milton's angels, focusing on the common held belief in Milton's age that angels were corporeal.
XVI focuses on Adam and Eve, and attempts to overcome the assumption that they would be naive, rather than fully formed and knowledgeable (though not fallen) human beings. XVII takes their relationship further and discusses the monumental task that Milton took upon himself by portraying "Unfallen Sexuality" in Paradise Lost.
Lewis briefly touches on the Fall in XVIII, and then offers a general review of Paradise Lost in the final chapter. Here he offers a brief look at the things he thinks Milton did properly, as well as the mistakes he made in writing Paradise Lost.
Let me first say that as someone who knows nothing about Paradise Lost, or even epic poetry, this was an interesting, engaging, and approachable book. I never once felt lost …- Aug. 27, 2010
C.S. Lewis the Last Line of in A Preface to Paradise Lost