Speak in an increasingly enthusiastic and poetic manner.
'Waxing poetic' has nothing to do with bees, candles, or polishing cars. The verb 'to wax' is 'to grow'; the opposite of 'to wane', which is 'to decrease'. Grow and decrease have largely superseded the archaic terms wax and wane in almost all modern usages, apart from the waxing and waning of the moon. The other remaining contemporary uses of 'wax' with the meaning of 'grow', survive in various expressions like 'wax poetic' and 'wax lyrical'. These are often explained as deriving from the imagery of the waxing of the moon. In fact, the word is extremely ancient and was used to mean grow in many contexts prior to it being used to describe the monthly increase in size of the visible moon. King Alfred, in the translation of Pope Gregory's Pastoral Care, which he commissioned in AD 897, used the Old English version of the word - 'weaxan'.
There are numerous examples of the use of 'wax', meaning 'grow', in mediaeval texts; for example, The Geneva Bible, 1560, in Deuteronomy 32:15:
"But he that shulde haue bene vpright, when he waxed fat, spurned with his hele."
[the 1611 version has it in more modern English as "Jesurun waxed fat, and kicked."]
It isn't until much later that 'wax' began to be used to refer to flowery and poetic speech or writing. This occurs in various phrases, like 'wax lyrical', 'wax poetic' and 'wax eloquent'. Of these, it is 'wax poetic' that is still most commonly used. 'Wax eloquent' was the first of this group of phrases to be used to describe someone becoming increasingly expansive and expressive in speech. That dates from the early 19th century, for example, this piece from Bracebridge Hall, a collection of essays and literary sketches by Washington Irving, 1824:
"The whole country is covered with manufacturing towns... a region of fire; reeking with coal-pits, and furnaces, and smelting-houses, vomiting forth flames and smoke." The squire is apt to wax eloquent on such themes.
Ironically, far from 'waxing eloquent', Irving was suffering from writer's block in 1824, following a family bereavement, and struggled to finish enough essays as to be worth publishing.
'Waxing poetic' came next. The first example that I can find in print is in Sir Henry Morton Stanley’s How I Found Livingstone, 1872:
"One could almost wax poetic, but we will keep such ambitious ideas for a future day."
Stanley seems to have been an enthusiastic waxer. His book also contains "I waxed indignant", "Farquhar and Shaw waxed too wroth", "I accordingly waxed courageous" - all at a time when he reports that the sun "waxed hotter and hotter". It would be remiss to leave out a Marx Brothers gag at this point. Groucho Marx's role as Professor Quincy Adams Wagstaff in the 1932 film Horse Feathers yielded this gem:
Wagstaff's Receptionist: The Dean is furious! He's waxing wroth!
Prof. Wagstaff: Is Roth out there, too? Tell Roth to wax the Dean for awhile.
'Wax lyrical' followed in the early 20th century; for example, Gilbert Cannan's translation of Jean-Christophe in Paris, 1911:
"He had the genius of taste except at certain moments when the Massenet slumbering in the heart of every Frenchman awoke and waxed lyrical."
Time for me to wane lyrical and stop.