I'm filling out an internship application, but it asks me why I should be chosen to become a Leader in their team (its a leadership internship),...

You are on the right track on your application in terms of thinking of specific examples. However, you might want to consider the larger picture.

What qualities of leadership do you have? Your extracurricular activities and jobs can be used described as assets if you think of your experience with them in leadership terms. For example, you might be able to say, "I'm a team player." "I learned about the importance of working together as a team in band, cheer, and on the basketball court." Then, you could give examples. This may or may not be true of you, but I am giving you an example of a leadership trait to help you think in terms of your strengths. 

One could think of many leadership traits learned in sports such as: strategy, hard work/grit, knowing the importance of preparation. This may help you think more deeply about your work. Ultimately, it's a personal answer. Remember these tips:

1) You want to describe what qualities you have that might make you stand out from the crowd. You always want to back up these assertions with examples. 

2) You want to talk about how your leadership qualities relate to the particular program to which you are applying.

Good luck on your application.