An important thing to consider (particularly if you are American, because we don't hear this that often) is that the colonists definitely still considered themselves British subjects. They saw their fight for representation as one against a tyrannical government, not a foreign power. To effectively look at why that is so, we need to look back far further in history than the American Revolution.

Certain British historians (and propagandists) like to think of all post-Norman English history as one two opposing forces: the decentralized Anglo-Saxon tradition of property rights and rule of law, and the highly centralized continental monarchial system imposed by the Norman invaders. Medieval England was a shockingly modern for an essentially feudal society. The nobility had a great deal of control, the title of King of England was not even an entirely hereditary position until the Norman conquest.

Basically, the Colonists were chafing under what they thought was unfair rule and looked at history as a guide. Many colonists were descended from people who fought in the English Civil War, and loyalist/patriot families were often descendants of cavalier/roundhead families in those days. The colonists subscribed to this notion of oppositional British history, and they were going to be damned if, after 600 odd years of resistance to corrupt monarchs and rule without consent of the governed, they were going to be the generation that finally let their rights as Englishmen be tossed aside. The colonists didn't see themselves as forward thinking radicals like the French Revolutionaries did a decade later, they saw themselves as conservatives defending Ancient rights that their ancestors had spilled to secure.

The English were much more sympathetic to the American cause than you may think. Edmund Burke, a premier statesman and the widely regarded as the founder of modern conservative thinking, was very sympathetic to the Americans. Decorated officers who had fought in the 7 Years War flat out refused to take up arms against the Americans. The British had such a hard time fighting the war that King George had to pay Germans to fight for him, which only served to anger the colonists even more. Eventually, after the first few years, people realized that even if the British won there was no chance of the same level of integration with the colonies as had occurred before, giving rise to the two separate countries we see today.

tl;dr the Americans saw themselves not as fighting for independence, but as resisting an increasingly tyrannical government in London and restoring the ancient rights of Englishmen to their proper place in the law.

I've skipped over a loooot. This is an incredibly complex issue and this is just one of many viewpoints, I just think that it's a fascinating way to look at one of the most misunderstood (and important) events in modern history.

For more information on this particular viewpoint, I would strongly encourage anyone to read Daniel Hanaan's Inventing Freedom: How the English Speaking Peoples Created the Modern World which is a bit chauvinistic at times, but gives a good overview of how the American Revolution can be seen as an existential crisis for essentially similar peoples rather than a genuine war of independence. Also, if you want to go deeper, reading relevant Parliamentary debates is highly encouraged, particularly from Edmund Burke or Charles James Fox, the two great proponents of the American cause in Parliament at the time.

Theodre Draper's "A struggle for Power", is a fantastic book on the American Revolution and is my source here.

America had been a very wealthy place for a while, and while the rich in America weren't as rich as the super rich in London, the average American was better off than your average Briton. The previous wars have proven that America can fight for itself, and now has no need for Britain. For YEARS British intellectuals have recognized that they need to keep them dependent by keeping them fearful of French aggression. Universities in the Americas also cut off some of the cultural connection Britain and America had. America not only didn't need Britain anymore, but were a prized possession of Britain, yet were still treated like nothing. This fueled feelings of nationalism in America, and the rich of America contributed to this because being independent would serve them better.

Plainly it was about many things. Contemporaries referred to it as both a war of independence, and as a revolution (see Chief Justice John Marshall's biography of George Washington). One easily forgets that war broke out when Great Britain attacked its own colony, occupied Boston. For remarkable evocation of the chaos and rebellion the British attack on Boston created, see Jill Lerore, Book of Ages: The Life and Opinions of Jane Franklin (2013). The Declaration of Independence followed the outbreak of war. All of the other strands of history mentioned in other threads undoubtedly contributed.

As to the revolution, the change in form of government, there is no consensus, and certainly no single reason. I choose, "all of the above." Modern accounts are dominated by Charles Beard, An Economic Interpretation of the Constitution, and Bernard Bailyn's masterly response,The Ideological Origins of the American Revolution (1967). The protection of slavery certainly played a part.

How was it about slavery?