One of the themes in it is how listening is a skill and applying it is one of the most powerful ways to influence people because it conveys to people, "I have an understanding of your concerns. When I advocate a course of action, I've done the work to take these concerns into account and evaluate trade-offs with them. You can trust me on this."
Books which also discuss the skill of listening include:
* Difficult Conversations, by the Harvard Negotiation Project
* Thanks for the Feedback, by the Harvard Negotiation Project
* and a bit of Decisive, by Chip & Dan Heath
Though the other things it says about decision-making processes are likely more useful, since persuasion involves influencing decision-making
Lastly, because both listening and persuasion both kinda depend on forming a good Theory of Mind, I'll recommend Thinking Fast and Slow. It is a bit farther-removed from your question, but also very generally useful.