The author just quotes an old phrase without knowing the origin or meaning of the verb. It is/was the opposite of wane.

The OED and Shorter Oxford definitions explain the relevant sense of the verb: To start talking moderately, then gradually increase in potency or intensity. Quoting the OED's typically long entry: "Originally a more frequent synonym of GROW v. , which has now superseded it in general colloquial use, exc. with reference to the moon [...] confined to literary use, and have, in varying degrees, a somewhat archaic flavour; [...] The verb is said still to be current in certain dialects"

While I have both of those dictionaries, the Shorter Oxford is the one I use and recommend.

"Waxing poetic" is an idiom meaning "talking longwindedly" or "droning on"; "wax" is used to describe talking, but you'll not find it used that way anywhere else.

The moon's phases are referred to as waxing and waning, wax being the same meaning in both cases: to grow.