The Declaration of Independence views the source of government and legitimacy as ultimately grounded within God-given natural rights. Jefferson famously 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.
As he goes on to write in the very next sentence, governments are created to protect those natural rights, "deriving their just powers from the consent of the governed." In short, government is created for a purpose (to safeguard the rights and freedoms of its people), and it only retains its legitimacy if it continues to honor that responsibility.
In essence, the American Revolution was founded on a political vision by which sovereignty was founded within the people and by which government should be answerable to the people. Of course, this should not be understood as an invitation to lawlessness on the part of individuals. Rather, real power resides within the people as a collective, who voluntarily submit themselves to the government in order to protect their own well-being.
So long as government continues to act for the benefit and well-being of those that it governs (and with the support of those it governs), then it would continue to hold legitimacy. Should it stray from that path and act tyrannically against its citizens, then the people are within their rights to overthrow it and institute a new government in its place. This is precisely the charge that the Declaration of Independence levies against Great Britain, and the logic that it uses to justify the American Revolution.