First, we must realize that not all of the American colonists did decide to become independent. Some of them actively supported Britain even after the war started. Others really didn’t care. That said, the American colonists who decided to separate from Great Britain did so because they wanted to have more control over their own government and economy.
Before the French and Indian War, the colonists had more or less been left alone to govern themselves. After that war, things changed. The British government tried to reassert itself and have more political and economic control over the colonies. By this time, the colonists were used to ruling themselves and felt that they had the right to continue to do so. When the British did things like imposing taxes on them, they became very angry. They felt that their rights were being trampled upon. They tried political means to get the government to let them be more autonomous. When those efforts did not work, they decided that they needed to actually break away from Britain rather than continuing to try to get more autonomy within the British system.
As an English teacher, I am going to answer this by providing a literary answer. I teach 11th grade, and therein my classes watch and read the play/musical 1776. If you can find a copy, I strongly suggest reading the appendix. It provides a detailed breakdown of the facts and fabrications detailed within the play (those instances where they adhered faithfully to historical fact and those instances where they strayed--most importantly, they explain why). Within the play it talks about the the mishandling of the colonies by the British crown, and how, in essence, the colonies (its people, resources, industry, etc) were used by the crown for its own benefit, but without receiving the rights, privileges, etc, common to any British subject. There is a really good line where the character of Ben Franklin offers that he would not mind being referred to as an Englishman were he given the full rights of an Englishman. Anyway, it is a wonderful resource that will allow you into the minds of these men and will bring history to life for you. Below are some links for you.