"To persuade the audience" means to convince or influence the people who are listening or reading something to believe or take action based on your ideas or arguments. Full definition
Attorneys can more accurately communicate and persuade audiences through routine practice and feedback. (lawyerist.com)
As for the relation between «identification» and «persuasion»: we might well keep it in mind that a speaker persuades an audience by the use of stylistic identifications; his act of persuasion may be for the purpose of causing the audience to identify itself with the speaker's interests; and the speaker draws on identification of interests to establish rapport between himself and his audience. (religion-online.org)
«Disorders of the voice can affect a politician's success: Politicians use their voices to persuade audiences of their leadership qualities, but how vocal disorders affect their powers of persuasion and listeners» voting preferences isn't so clear.» (sciencedaily.com)