Why do some fellowships require "leadership experience"?

Precisely because we cooperate a lot, we need good leadership. A good leader is someone who can bring the best of everyone in the group. For example, encouraging everyone to voice their ideas, assessing which ones have actual value, but without making the group pursue all the wild geese. A good leader should also be able to recognise the strong points in the members of the team and assign tasks accordingly.

I accept that everyone here should be motivated, but it is important for the group to keep it up. No matter how eager I am, if my professor were to start giving me contradictory orders, unreasonable workloads, or dismissing my ideas without explaining why, I can tell you, I would not remain that motivated after four years. After all, I can be motivated to work "in the grand scheme of sciency things", but not necessarily in the particular group I am in.

The description of this particular fellowship includes the phrase:

Winners are chosen based on individual merit, including academic performance and preparation, intellectual capability, English proficiency, and the potential for contribution to scientific education and research in Vietnam.

"Leadership" means the ability to work with and through others. It is much harder than people think - a good leader is humble yet decisive, a good listener yet able to motivate; he can synthesize the thoughts of the group into a common goal and vision, use that vision to obtain resources and allocate them in a way that helps the group achieve its goals. Doing all that without appearing to be "the boss" is real leadership - something that comes with practice. Selling cookies to support your local charity is initiative; getting together with your friends to sell lots of cookies and build a new school, that's leadership. I have heard it said that

Leadership is what bridges the gap between responsibility and authority

Leadership lets you change the scale of your impact; and since this particular fellowship is explicitly created to find individuals who will have impact on scientific research and education in Vietnam, you need people who have both the academic skills and the skill to translate this into impact "on the system".

Demonstrated leadership experience is an opportunity for the selection committee to explore whether you will be able to make an impact - they are not looking for the next CEO, but you will be amazed how much difference a good leader can make in any collaborative environment.

In the US this trait is becoming so highly valued that some high schools have an explicit course "Leadership" on their curriculum - a chance for students to develop and practice these kinds of skills, often in the context of community projects.

It is obvious from the selection summary that intellectual ability, preparation etc. are most important - but I hope you can see that leadership as I tried to define it here has a place in this academic environment.

Scientific breakthroughs are rarely achieved by a single scientist, but by a research team. It is important that the team leader can identify relevant research questions and direct the group members' work to use suitable approaches to solve them. Often, the breakthrough results are associated more to the group leader than to any group member, even though the group members may have done 80 to 99 % of the actual research work.

Fellowship funders try to look into the future of your career. You will only stay in academia after the PhD or maybe Postdoc level if you have the ability to lead a research group. That is, apart from rare exceptions, you will only be able to associate big scientific results with your name if you have leadership abilities. And these results are what funders try to support in the end.