There are several ways in which Lincoln manages to connect the Gettysburg Address to the Declaration of Independence. For example, in the first paragraph, Lincoln tells us that:
"Four score and seven years ago our fathers brought forth on this continent, a new nation, conceived in Liberty, and dedicated to the proposition that all men are created equal."
This passage reminds us of the fact that liberty should be found at the core of the United States. Moreover, it also supports the idea of the equality of men, which is mentioned in the Declaration of Independence.
Another example occurs in the last lines of the speech. Lincoln states that:
"... this nation, under God, shall have a new birth of freedom—and that government of the people, by the people, for the people, shall not perish from the earth."
This passage mentions that the government is created and controlled by the people, and that it has a responsibility to address their needs. This is an idea that was also mentioned in the Declaration of Independence.
The main reason why Lincoln decides to link these two texts in such a way is because he wants to remind people of the origins of law and government in the country. Moreover, by linking both texts, he makes his argument more credible.