The colonists pinned the legitimacy of their rebellion on the concept of "natural law." This conflicted with British monarchial thinking, which said the colonists needed to obey their king because the king was appointed by God to rule the people. Power was top-down, and to oppose a king was to oppose God's will.
Natural law, in contrast, stated that all humans are born with inherent rights, rights granted them directly by God and which cannot be taken away from them. These rights will sound very much like what we read in the opening of the Declaration of Independence, and with good reason, as the colonists were aggressively establishing an alternative reading of God's will for the people. Looking at Enlightenment thinkers such as Locke, they were asserting that kings had no right to rule without the consent of the governed and that people had a God-given right to revolt against tyranny. As the Declaration states:
We hold these truths to be self-evident, 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 . . .
We can skim over these over-familiar words, but each one is carefully chosen to stake out a position: the colonists are asserting that they have rights that come directly from God ("their Creator") and which are "inalienable" (can't be taken away). Government exists for the good of the people ("the governed"), not the good of the king, and people have a "right" to overthrow governments that don't attend to the fundamental liberties and needs of the people.
These are fighting words and, because we accept a republican (democratically elected, non-monarchial) form of government as natural, it is easy for Americans to lose sight of how radical these concepts were then. The more conservative British could have leaned into the Bible and countered that the king had every right as God's representative on earth to use an "iron rod" to break a stiff-necked people, who had not the least business daring to rebel. A republic was considered by many as breaking away from the way God meant the world to be ordered and an invitation to chaos. Luckily, the colonists could fall back on alternative, Enlightenment, philosophies.