Different people, indeed different educational institutions, will give you different answers to this. I will first cite a very widely-read and appreciated speech on the topic delivered by historian Peter Stearns to the American Historical Association. It is often used by teachers as an introductory text to history courses, and is passionate in its vindication of the usefulness of history. Stearns argues that the study of history is important for two fundamental reasons. First, it "helps us understand people and societies." By this, Stearns means that even social scientists must use history to make predictions, since we can't really conduct controlled experiments with events as they happen in the real world. Second, Stearns says that "history helps us understand change and how the society we live in came to be":
Only through studying history can we grasp how things change; only through history can we begin to comprehend the factors that cause change; and only through history can we understand what elements of an institution or a society persist despite change.
Historians track and interpret changing trends ranging from alcoholism to voter turnout rates, Stearns suggests, and our understanding of how these phenomena, which are obviously relevant to our present lives, have historically functioned can inform our responses to them in the present.
Additionally, Stearns argues, history provides a sense of identity and can help shape our understanding of morality. The study of history also fosters important skills, including interpreting evidence, critical reading and writing skills, and argumentation.
In addition, history helps foster another habit of mind that is unique to the discipline. History has what a National History Center report has described as "an analytical imperative to step outside oneself," in other words, to try to understand events through the point of view of people who lived in a different world than our own. This sense of empathy requires a tremendous amount of intellectual discipline, in fact, educational psychologist Sam Wineburg has called it an "unnatural act." As noted above, it is more fundamental to the study of history than any other discipline, because history alone requires us to reach across time.
Sources: Samuel Wineburg, Historical Thinking and Other Unnatural Acts: Charting the Future of Teaching the Past (Philadelphia: Temple University Press, 2001).
The National History Center, “The History Major and Liberal Education,” Liberal Education 95: 2 (Spring 2009) 42.