When you flesh out an argument or an idea, you add more details to it, giving it more substance. Your English teacher might ask you to flesh out the rough draft of your essay.
While it's possible to flesh out a speech or a conversation, it's more common to find this phrase referring to written work. A poet's editor might tell her to flesh out a certain section of a poem, or you might flesh out the core of your argument in a story you hope to publish in a magazine. Flesh out is believed to come from the idea of adding flesh, physical substance, to a skeleton or a frame.