Why do the signers of the Declaration say that it is their duty as well as their right to alter their government?
The signers, as free citizens, determined that their unalienable rights are "Life, Liberty, and the Pursuit of Happiness." The government is instituted to preserve these rights. If the government infringes or takes away these rights, it is the right of the people to change or replace the government so that those rights are protected. In a roundabout way, this is to say that it is the right of the American citizen to enjoy these three rights. Or, less redundantly, American citizens would be justified in altering their government if that government limited or withheld these rights.
Each citizen, being equal, has the right to live, be free, and pursue happiness. (Since slavery was still legal and women could not vote, clearly not all citizens were equal at the time of this signing, but for the sake of argument, let's analyze the statements themselves.)
Since all citizens are equal under this declaration, each and every citizen absolutely deserves these rights. So, if some citizens are not getting these rights (for whatever reason), it is up to all citizens (those who enjoy those rights and those who do not) to protest and/or alter the government so that those rights are reestablished. The signers set the standards that all citizens (well, it is "men" in the declaration; but for the sake of modernizing the argument, I am using "citizens") . . . that all citizens are equal. If even one citizen (who is equal to everyone else) is denied these rights and freedoms, then it is the duty of all citizens to challenge or alter the government so that citizen enjoys the same rights as everyone else. It is their "duty" because the signers established that the government must be altered if any of their equal citizens are denied these unalienable rights. More colloquially speaking, "I am entitled to these rights and so are you because we are equal under the law. Since we abide by this code of equality and freedom, we are both obligated to alter the government if either of us (or anyone else) is denied these rights.
The Declaration of Independence was written to present the reasons why the colonies were taking the unprecedented action of revolting from the ruling governmental power (Great Britain) and creating a new country. The first paragraph affirms the responsibility of the colonists to "declare the causes which impel them to the separation."
The second paragraph sets out a list of basic human rights that the authors of the Declaration contend are "self-evident" and that need to be granted to all persons. These rights include the equality of all men, the possession of "certain unalienable rights" given to all by God, that governments are created "to secure these rights... deriving their just powers from the consent of the governed." It then goes on to present the argument that when a government ceases to support and protect those basic rights, it becomes "the Right of the People to alter or to abolish it, and to institute new Government."
The Declaration emphasizes that this is not a step to be taken lightly. It affirms that people will generally be willing to continue to exist under a familiar governmental situation rather than make such a radical change. However, the Declaration contends that when a government has established a long pattern of "abuses and usurpations" that are taking away the basic rights of the people, then "it is (the peoples') right, it is their duty, to throw off such Government, and to provide new Guards for their future security."
The Declaration of Independence was the formal statement to the English and to the world that the colonists were exercising their duty to throw off the abusive English government and establish a new government that would be more respecting of their rights.