When you pull strings, you use your insider status to get something you want. If your college application gets a second look because your mom knows the admissions director, you pull strings to increase your chances of acceptance.
This idiom has been around since the 1800s. It comes from puppetry and the way a puppeteer pulls a puppet's strings to make it move and perform a certain way. When you pull strings, you use your influence to make people do things to your advantage, whether it's asking your uncle who owns a candy store to hire your best friend or leveraging your family's friendship with a theater director to get yourself a starring role in her new play.