The Declaration of Independence is not a list of rights but was rather a declaration of war. The legal rights of the American people would not be officially put on paper until fifteen years later with the ratification of the Bill of Rights in 1791.
This is not to say that the Declaration of Independence does not mention any rights. Its second paragraph spells out the lofty Enlightenment-era notions to which its writers, namely Thomas Jefferson, subscribed. Philosophers such as John Locke, Montesquieu, and Voltaire had argued that a government's primary responsibility was to protects a person's natural rights. These rights, as spelled out in the Declaration of Independence are life, liberty, and the pursuit of happiness. These three rights are taken directly from a work of John Locke's titled Two Treatises of Government. Jefferson, quite the philosopher himself, agreed with Locke that all people have these rights and no legitimate government could deny them to the population. (Of course, it would be generations before women and non-white citizens would actually secure these rights.)
Since, as the Declaration of Independence argues, the English government was denying these rights to the colonists, it was their duty to overthrow the old government and institute their own in order to protect these rights.