The answer to this question can be found at the end of the Declaration of Independence. It is at that point that the signers actually come around and say that they are breaking away from England. Before that, the document had been giving reasons why the colonists would be justified in breaking away, but it never actually said that they were. Towards the end, the signers explicitly state that they are breaking away and they say that they are doing so by the authority of the people of the American colonies.
In the Declaration of Independence, Thomas Jefferson (who did most of the work of writing the document) emphasizes the importance of democracy. He says, among other things, that government can only have legitimate authority if the people consent to be ruled by that government. If Jefferson is going to say that democracy is important, he needs to appeal to democracy when he says that the colonies are breaking away from England. He has to say that the people approve of this step or else the step will not be democratic and will be illegitimate. This is why Jefferson says that
We, therefore, … do, in the Name, and by Authority of the good People of these Colonies, solemnly publish and declare, That these United Colonies are, and of Right ought to be Free and Independent States…
Clearly, then, the representatives who signed the Declaration of Independence are claiming that the authority of the people of the colonies is what gives them the right to declare independence.