What does wax lyrical mean?

Answer 1

It means to talk positively about someone or something.

The intransitive verb to wax is normally seen in the sense of increasing, as the moon waxes and wanes in its cycle. But to wax also means to become, to tend towards, to grow towards. One can wax indignant, or wax lyrical, or wax rapturous, meaning you become/tend to that state. The word is old, tracing back from the high German to the Greek.

"Wax lyrical" is an idiom that means "to begin speaking or writing in a poetic or musical style". (one meaning of "wax" is "to pass from one state to another")

1. (idiomatic, transitive) To talk about something with much interest or excitement.
2. (literally, transitive) To become, or tend to become lyrical.

Answer 2

It means starting to speak poetically or talk enthusiastically and constantly about a subject or experience.