This phrase is still fairly well used — ‘waxing poetic’ in the US and ‘waxing lyrical’ in most other English languages. The fact that waxing lyrical and waxing poetic both exist and are deemed interchangeable makes me think it is related to the bardic tradition.
Both phrases are used when someone grows particularly loquacious and flowery in their communication and nothing summarises the role of a bard quite as well as loquacious communication. A bard’s job was to sing the history and politics of his people to the accompaniment of a lyre or lute and to ‘sell it’ with engaging, inspiring eloquence. They were called poets though their works were seldom written down and everything was done verbally. But in the same way that it is the job of poetry to capture or comment on an aspect of human nature, so bards captured and commented with their poems set to music.
‘Wax’ is from an Indo-European word meaning ‘to increase’ or ‘to grow’. So the bright half of the month is known as the ‘waxing moon’ when it starts to fill out and grow.
The latin lyricus (from which lyrical comes) means ‘of or for the lyre’ so this was something to be communicated on a lyre. So while (later) poetry came to be spoken and written, back in the day it was sung to a listening audience and composed with the same rigorous structural and cultural traditions as other poetical forms.
So ‘waxing lyrical’ means you grow poetic and over-the-top in your language and perhaps it was meant to imply that you have grown as loquacious as a bard whose very survival relied on their ability to sing a fantastic and evocative tale.