What does it mean to be a global citizen?
A citizen of the world, or cosmopolitan, is someone who values the equality of all people regardless of where they're born. Where the modern civil rights project has worked to provide for the civil, legal, and social equality of people regardless of race, sex, class, religion, sexual preference, disability, and more, a cosmopolitan wants to say that there's another group of people who are just as equal as everyone else: all the people who live or were born outside the borders of their state. People aren't morally responsible for the place in which they're born any more than they're responsible for their race, how tall they are, or whether they can jump up and down on one foot while singing "Row Your Boat." We all belong to one vast community of humans in addition to each of the smaller communities we belong to, and that most populated community matters just as much as our families, neighborhoods, cities, provinces, and nations.
The central tension here, though, is between cosmopolitanism and nationalism, or in a multi-national state like the United States, as a tension between cosmopolitanism and patriotism. Rather than owing special respect to our fellow citizens in the nation-state, the cosmopolitan believes that moral worth belongs equally to all, regardless of nation, state, or ethnicity of origin. In global politics since the 17th century, the main unit of concern has been the nation-state, so saying that you care about individuals at the global level is somewhat radical.