Typically, a question like this would be asking why it was that the American colonies chose to become independent from the United Kingdom. However, I do not think that this can be the meaning of the question when it relates to Letters from an American Farmer. These essays are really not about the Revolutionary War, but more about what it meant to be an American in the years just before that war.
Most of these essays were written in the years before the war. They are not really focused on the politics of the Revolution. We do see Crevecoeur write in Letter XII about the fighting, but he does not talk much about the reasons for the war. In fact, he does not seem to put much store in what the leaders of either side say about the war. As he says,
The great moving principles which actuate both parties are much hid from vulgar eyes, like mine; nothing but the plausible and the probable are offered to our contemplation.
What this means is that the two sides only tell the public what they think will sound good. The two sides do not really tell the truth. This implies that Crevecoeur has no real views on which side is in the right or even about what the real causes of the Revolution are.
So, how to answer this question? I would say that it is asking us why Americans became more emotionally separate. I would therefore use Letter III (which is also the most famous of the letters) to answer the question. I would say that Americans separated themselves (emotionally and mentally) from Europe because Europe did not do them any good. Crevecoeur asks us
What attachment can a poor European emigrant have for a country where he had nothing? The knowledge of the language, the love of a few kindred as poor as himself, were the only cords that tied him: his country is now that which gives him land, bread, protection, and consequence…
This is what I would use to answer this question. Here, he is saying that immigrants to the colonies stopped identifying with their native lands because those countries never gave them “land, bread, protection, and consequence.” In other words, people felt like they meant nothing to their original countries. Because they meant nothing, they separated themselves from those countries (physically and emotionally) and came to the American colonies.