You're asking the big question. Why is it important? To us historians it seems obvious that it is, but it can be hard to put under words.

A little exercise

One way to try to explain it would be the following exercise: Think about yourself and who you are. What is important about you? What defines you? Take a moment to think about this; perhaps write a little list on a piece of paper. Now imagine you know nothing about what happened before today. All that there is is you, as you are today, with no memories, no previous achievements, nothing. Go through the list you made earlier. How many things wouldn't be there? If the list is on paper, scratch out those things. What remains? For many people, the answer will be "not a lot". Your personal history has made you who you are. Similarly our shared history has made us (our community, our country, our world) the way it is. If we forget our history, we forget who we are.

A more fundamental response

Lots of people have thought about why learning history is important. They come to slightly differing conclusions, but they all come to the conclusion that is important. I'll list some different reasons why learning history is important below; the relative importance of these reasons can be debated, but all have been put forth as being of some importance:

• History provides insight. - History allows people to find out whether or not something actually happened, and how we know that it did (or didn't). The is where the historical method is important.
• History lets you think about what's good and what's evil, what's effective and what's ineffective. While historians try not to judge the past, history allows people to compare current issues with issues in the past. If something bad is happening, how have people in the past reacted to it? Have individuals or organisations been able to influence events for the better? If so, how?
• History can inspire. This inspiration can be great or small. Great deeds by great men and women of the past can inspire people today to try to achieve great things. But also the smaller things can provide inspiration. A photo of a peace activist putting a flower in a rifle can inspire people today to be more peace-loving.
• History lets people think critically. This is somewhat related to the first point I mentioned (how do we know that something happened), but it's also more broad. If one realises that a historical source can give a false image of what happened, one might realise that current books, papers, images may not provide an entirely accurate view.
• History makes people politically involved. This is not really a goal many historians like to admit, but for many decades this has been the explicit goal of history education in many countries: to make "good citizens" out of students, to make them love their nation-state.
• History leads to self-knowledge. This is related to the exercise I started this post with. Without history, how can you know who you are? How can we know who we are?
• History leads to contemplation. Studying history often leads to big questions such as: Why is there conflict in the world? Why or some countries more prosperous than others? How are societies shaped? This kind of fundamental, almost philosophical, questions lead to a better understanding of the world.

Note that this is not an exhaustive list, nor is it a list which all historians would agree with. The points above have been adapted from Glöckel, Geschichtsunterricht (1973) and have been commented on, elaborated on, built upon many times since. Unfortunately I'm unable to elaborate further at this time. But the post is long enough already and I hope you have gained some arguments to bring to your friend :)

I've always said that nobody ever killed each other over what 2 + 2 is equal to, but people have been killing each other for a very long time over who Jerusalem belongs to. Studying maths and science is important, but it should not be at the expense of history, because studying history is the key perhaps not to solving the world's problems, but it can do something to prevent future problems. Israel is always in the news, but do you understand why? What decisions led to putting a state of Jews in the middle of Arabic Muslim-controlled territory? Or take the Syrians - why are people so nervous about Russian intervention in the war? It's because these things similar to this have happened in the past, with well-known consequences. You can understand why a government works the way it does based on what happened in the past to shape it currently. You can understand what past governments did wrong - such as why a government's spending policies might spark a rebellion(French Revolution) or why a dictator like Hitler came to power, even after he outlined pretty much everything he intended to do in Mein Kampf for the world to see.

At the very least, next time you are watching the news, you might be able to put into context a news story about the refugee crisis or American presidents. You might actually know what you are talking about next time you give your opinion.

There are whole modules on degree courses focussed on why we study history, it is very difficult to explain.

We should study, and continue to study history because it is always changing. There is no master narrative from which all others stem.

History itself is never constant, for example; just a decades ago events like the English Civil War were primarily considered to be about political/economic issues (Structuralist view, C. Hill ect. ect.). However now historians (at least most of them, the Cultural turn) have changed their mind, now if you were to read an article or book on the Civil War you would primarily see religious motivations displayed, it is even being described as the last of the wars of religion now. (and that is not even talking about Cromwell in Ireland, we are talking about 1642 here!)

With the new (in the grand scheme of things) Cultural turn what we typically study in history has changed. For example we have gone away from writing 'the history of' something to 'a history of' something. Cultural history is a massive definition which seems to include most things so I'll give an example.

example 1 Composition of Armies in English Civil War (historicist) Biographies of Generals and Leaders

example 2 (same question) (Structuralist) Demographics of armies, socio-political groups from which ppl come from

example 3 (same question again) (Cultural term) case study of some random minor party, or, what were common people fighting for, or, investigating make up of armies through petitions/pamphlets/propaganda (or something like that)

Each history is about the same topic, but the approaches to study that topic are very different. Are all the approaches equally valid, well if we started that debate we'd be here all day.

People here have said some very valid and powerful points, but I also like to add that learning history means that no one person can just shape history for his uses. The holocaust is a prime example. It is probably one of the most well documented massacres in history, but still you get people, and even countries denying it ever happened or nearly to the extent it is recognized in the world.

Try to imagine what it can be like if a man can shape what you know of history and make the Nazis look like the saints trying to save jews from all of that oppression in eastern Europe and giving them shelter in Poland.

Past events effect todays reality, it affects peoples motivations and what they believe they are owed. When people are completely ignorant of that, they are easily manipulated to believing whatever is told them, no matter how absurd it might be.

u/polybios covered it for the most part. I would add that it teaches how great men dealt with adversity and how lesser men failed to do so.