To "abuse" a power or a privilege means to use it to do bad, selfish things. In this case, it means to use your authority in ways that benefit you unfairly, or in ways that hurt other people. For example, here are some examples of people "abusing their authority":

• The admissions director for a university lets her nephew into the school, even though he didn't have good grades in high school.
• The head of a department makes his employees go out to get coffee for him in the morning.
• A junior high school soccer coach sometimes yells at the players until they cry.