What is the meaning of waxing eloquent?

It means to speak with eloquence (in this context, "eloquence" is the art of using language with fluency and aptness). Overall, "waxing eloquent" means to make a speech using nice Words. The term is often used in areas where the topic at hand considered controversial, yet the speaker able to pull it off to get message through without insulting others. "He waxed eloquent when speaking out for the citizens against that multi-national corporation allegedly causing cancer clusters of local citizens".