I suppose the first question you would have to ask yourself is "what does it mean to be a 'better global citizen?'" Technically the concept of being a global citizen means that you do away with traditional governments and have everyone be part of a one giant type of global country. I suspect that's not what your question is asking, though.
I suspect that your question is really asking about being a "global citizen" in terms of being aware of, and caring about, what goes on in the rest of the world. In this sense, global history becomes important in the same way that national and local history is important to us now. How can we understand the problems of the world, and work toward fixing them, unless we understand how those problems began in the first place? Studying global history gives people the chance to understand how we got ourselves in some of the messes we are in: starvation, war, environmental damage...you name it. Without understanding the history behind a region of the world it is impossible to suggest ways to move forward toward making the world a better place.
For example, we know that the rain forests are disappearing in countries like Brazil, but why? What forces are in play that make people destroy their environment by burning the rain forests down in the first place? Studying the history of that area and situation would give us clues as to why it happened and what we might be able to do in the future to fix it (or keep it from happening again somewhere else.)
The more you know about people the more likely you are to embrace them. Ignorance is the enemy of people being able to get along. The more you care about people who don't immediately live around you (being a good global citizen, that is) the more likely you are to care about their situations and want make the world a better place to help them.
I hope this helped!
I would say this is true no matter what country you are a citizen of, or which history you are studying. As a teacher of history, this is one of the things I tell my students each year. By studying and knowing the history of your own culture and country and that of the world, you come to understand your own identity, your own place within that world.
It also helps you to develop an appreciation and understanding of other cultures, which promotes tolerance and further study. One of the flaws in American society, I would argue, is that we too often look at the world through Amero-centric glasses, preferring to make decisions as individuals and society as if we are the whole world, instead of the 5% of it we represent.
The teaching and learning of history are insurance against that kind of nearsightedness.