How does word repetition and “refrains” (for example, “I have a dream today” or “Let freedom ring”) build persuasive power?

Forms of repetition have been acknowledged rhetorical devices used in speeches intended to persuade since the days of the Ancient Greeks. Simplistically, the value of repetition in persuasion is simple: the more a person hears a convincing phrase or point, the more likely they are to accept it and be convinced by it.
The example you cite, a refrain, plays into a wider rhetorical technique called “climax.” Where some repetition uses the “power of three” or anaphora across a paragraph to create intensity within a short space or on a particular point, prolonged use of a refrain can build climax over the course of an entire speech. Between each refrain, techniques such as enumeratio lend credence and power to the refrain: each time the listener hears it, there is more nuance connected to it and more power behind it.