There were many factors that led American colonists to rebel against the British, but the French and Indian War (1756–1763)—or the Seven Years War for Europeans—is a good starting point.
American colonists believed they played their part well in the fight against the French and the Indians. So when the British Parliament, and King George III, began imposing new laws and taxes (Sugar Act, Currency Act, Quartering Act, Stamp Act) on the colonists, to pay for the war and to reduce British debt, the colonists felt betrayed and proceeded to try to reduce the effect of these laws. They generally succeeded in doing so. But the British didn't quit, and they began to impose even more laws on the colonists.
Acts of resistance, like the Boston Tea Party, were small but significant gestures meant to demonstrate to the British that the colonists were resolved not to be taxed to pay off British debt. Moreover, there were no colonial representatives in the British Parliament representing colonial views on anything. Hence, the colonial response to any new British tax was "No taxation without representation!"
But during this time, the people in the thirteen colonies were not united, and they really had no plan to become united. Nevertheless, the British imposed a new set of laws, the Intolerable Acts, which included the Boston Port Act, that punished the colonists for their continued resistance.
The colonists had had enough as well, and in 1774, twelve of the thirteen colonies sent representatives to the First Continental Congress in Philadelphia, PA. They had hoped to reverse the effects of the sanctions imposed by the Intolerable Acts. Among other things that came out of that one-and-a-half month long conference was the decision to boycott the purchase of all British goods.
It was in the following year, however, that the "Shot Heard Round the World" set off hostilities between the British and the American Colonists on April 19, 1775 that began the American Revolutionary War.
As you can see, there were many factors that impacted the American Revolution, but the fact that the colonists kept rebelling and kept responding to the laws imposed on them is what eventually led to the beginning of the war.