To wax poetic means to become poetic (with one’s language). The verb to wax usually describes the moon developing towards fullness (full of light). The moon waxes from dark to full over a two-week period. Another verb, to wane, describes the moon going from full to dark. We say the moon is waxing / waning. Other things that wax and wane include sounds that grow louder and quieter and emotions that grow stronger or weaker:
The train’s noise waxed as it approached.
His anger waned after he received an apology.
We say that someone waxes poetic when their words become more poetic over a period of time:
As he described the mountains back home, he began to wax poetic.