First, a definition of conversational copywriting.
To me, it means writing in the same way you would speak persuasively to a friend over the kitchen table.
Imagine you’re trying to persuade a friend to go hiking with you at the weekend. You’re in conversation, but you’re trying to be persuasive as well.
Because it’s within the context of a real dialog between two people, your language is natural and conversational. Emotional at times.