How does allusion give authority to one's own ideas?
Allusions are references.
Dr. Martin Luther King Jr. frequently used allusions (such as ones to Augustine of Hippo, St. Thomas Aquinas, Meshach, Shadrach, and Abednego). Allusions are used to illustrate one's point. They can be used as examples by someone making a point.
Dr. Martin Luther King Jr. used quotes and allusions from well-known religious saints and authorities to give credit for his own work. It's like using another person's reputation for your own advantage.
In his "Letter from Birmingham Jail," King talks about how the Hungarian freedom fighters were considered illegal while Hitler's atrocious acts towards Jews, Gypsies, homosexuals, disabled peoples, etc. were legal in Germany. He uses this to showcase how injustice is never right even if the law supports it.
This is a powerful allusion that could convince others that his cause is morally correct. King uses this to his advantage.