David Lavenda wrote a nice blog post Once Upon A Time at the Office.  He offers some reasons for why stories work to persuade but also offers 10 tips for constructing a story.  His ten tips are good and if you are new to storytelling, very helpful, take a look

 http://www.fastcompany.com/3015140/leadership-now/once-upon-a-time-at-the-office-10-storytelling-tips-to-help-you-be-more-persu

But what was really helpful was the quote from Robert McKee, a screen writer's legend.  Mr. McKee is quoted as saying “Trying to convince people with logic is tough for two reasons. One is they are arguing with you in their heads while you are making your argument."  So simple, most people when in an argument/debate they do not listen well to what their opponent is saying they are formulating their response.

Stories break that cycle by engaging the listener in the story and taking him or her on a journey with the teller.  Together they can discover the truth.  Much easier to persuade since the person appears to reach the best decision on his own.