What does "waxed rhapsodic" mean?
To wax rhapsodic about something is a common idiom meaning to praise something excessively.
As James indicated, the verb to wax is a verb meaning to grow, but the verb is very uncommon outside of a few contexts such as this idiom, and when speaking of the phases of the moon.
In this context, "to wax" means "to grow." It is also commonly used to describe the moon as it goes from new to full.