Why did the colonists believe they were justified in breaking away from British rule?
Perhaps the best place to look for the answer to this question is the Declaration of Independence. It provides an explicit justification for breaking away from Great Britain (which, by the way, even many revolutionaries had been uncertain about beforehand.)
The Declaration says that:
...all men are created equal; that they are endowed by their Creator with certain unalienable rights; that among these are life, liberty, and the pursuit of happiness; that, to secure these rights, governments are instituted among men, deriving their just powers from the consent of the governed; that whenever any form of government becomes destructive of these ends, it is the right of the people to alter or to abolish it, and to institute new government, laying its foundation on such principles, and organizing its powers in such form, as to them shall seem most likely to effect their safety and happiness.
What this means, basically, is that all men are born with certain rights, and that the purpose of government is to protect those rights. The only legitimate government is one that preserves the rights of the people, and that is founded by their consent. If a government fails to protect the rights of the people, or are actually destructive of them, then the people have the right to replace it with another.
The American revolutionaries argued that they had been the victims of a "long train of abuses" that gave them the right to break away from British rule. These included taxing the colonists without their consent, closing their ports, dissolving colonial assemblies and refusing to approve their laws, placing troops among them, and, in their words, encouraging "merciless Indian savages" to make war upon them. To the colonists, these offenses more than justified breaking away from Great Britain. They also thought, correctly, that declaring independence would open the door to loans from European nations necessary to fight the war, and to an alliance with France. But the moral and political justifications for independence are found in the document they used to explain their cause to the world.