The narrator is a man who believes he has discovered an invasion of "nonhuman species of incredible properties" after finding a paperback book on a bus. He takes idioms literally instead of figuratively, which demonstrates the complexities of the English language. For example, when he reads about "eyes slowly rov[ing]" around a room, he believes that those eyes have become detached from the being and roll around "like dimes" independently. When he reads about a lady asking for a man to "remove" his arm from her, he believes that arms and other portions of anatomical features are detachable "at will."
In some ways, the narrator reflects an ordinary man. He has a family, owns a house, and watches his family playing Monopoly in their kitchen. Yet the narrator's inability to reconcile familiar metaphorical language indicates his seemingly sudden break with rational thought. After all, he has surely heard these types of idioms before but is suddenly convinced that these phrases indicate a sinister plan for "them" to "invade Earth."
On one level, the narrator is quite thoughtful. Indeed, most speakers use phrases such as "losing one's head" and "having no guts" without processing the phrases with any sense of reflection at all. There is an innate cognitive dissonance in many idioms speakers simply accept and use mindlessly. The narrator of this story is giving those phrases their due mental attention and has discovered that something is amiss in the English language.
On the other hand, it wouldn't be accurate to label him a truly thoughtful narrator; his leap from considering the inconsistencies of language to believing that those phrases indicate an impending invasion of a nonhuman species doesn't demonstrate the mind of a complex thinker.