The purpose of a persuasive speech is to convince the audience to adopt the speaker's perspective on a given topic. The core of a persuasive speech is pathos: appealing to and resonating with the audience's feelings and emotions.
In order for the pathos contained in a persuasive speech to be effective, the speaker has to understand the audience he/she is addressing. To be convincing, the speaker has to take into account the behavioral motivations and foundational beliefs of the audience. Showing empathy with the audience is crucial. Drawing parallels between yourself and the audience reduces the distance between you and them, making your speech that much more persuasive.
In addition to pathos, persuasive speeches contain appeals to ethos and logos. An appeal to ethos is used to show the character of the speaker and make him/her more credible. For the audience to be persuaded, they have to feel that the speaker is a credible and worth listening to.
An appeal to logos requires referencing evidence. This demonstrates the extent to which the speaker is knowledgeable about the topic he/she is speaking about, making their speech more persuasive than if he/she appeared ill-informed on the topic.
Source: Source: Boundless. "Persuasive Speeches." Boundless Communications Boundless, 27 Feb. 2017. Retrieved 26 Jun. 2017 from https://www.boundless.com/communications/textbooks/boundless-communications-textbook/introduction-to-public-speaking-1/types-of-public-speeches-22/persuasive-speeches-106-4201/