Lincoln is arguing in these lines that the continued existence of democracy is predicated on the abolishment of slavery.

According to Lincoln's Springfield law partner William Herndon and noted in the biography he wrote, Abraham Lincoln; The True Story of a Great Life, Vol. II, p. 65, that phrasing comes directly from a lecture by Theodore Parker, a minister from Massachusetts. Herndon sent transcript of the speech to Lincoln, and Lincoln returned it, expressed his thanks, and noted his appreciation for the line (which he marked in pencil on the transcript):

"Democracy is direct self-government, over all the people, for all the people, by all the people."

If we look directly at Parker's work, he uses the phrase in two different contexts.

Theodore Parker, "The Effect of Slavery on the American People" (speech given at The Music Hall, Boston, Massachusetts, July 4, 1858).

This first is near the beginning of the work and is addressing people's actions in a democracy as not only accountable to god, but also to their fellow man [emphasis mine here and below]:

Theocracy, the priest power, monarchy, the one-man power, and oligarchy, the few-men power, are three forms of vicarious government over the People, perhaps for them, not by them. Democracy is Direct Self-government, over all the people, for all the people, by all the people. Our institutions are democratic: theocratic, monarchic, oligarchic vicariousness is all gone. We have no divine vicar who is responsible to God for our polities and religion; only a human attorney, answerable to the people for his official work.

The second appears at the conclusion of the piece:

Nothing can save Slavery. It is destined to ruin. Once I thought it might end peacefully; now I think it must fall as so many another wickedness, in violence and blood. Slavery is in flagrant violation of the institutions of America—Direct Government,—over all the people, by all the people, for all the people. It is hostile to the interests of industrial Democracy; it lessens wealth— weakening the growth of creative power, Toil and Thought.

As the first quote doesn't have any great meaning here, I would argue that it is this second iteration of the text that we should be examining. Parker viewed slavery as an enemy of democracy and demon to a representative form of government. Lincoln is continuing that line of reasoning through his words in the Gettysburg Address.

The intention of his narration becomes clearer when we add in the previous line to Lincoln's speech:

"that we here highly resolve that these dead shall not have died in vain."

He here is again referencing the war as an end to slavery. He hopes that the war will end, and slavery - the threat to democracy - abolished thus giving the death of the soldiers at Gettysburg higher purpose.

It is noted quite frequently on here, and as nuanced as it was, the war was about slavery. Giving context and intentions to Lincoln's words simply reiterate that point.

Thanks for asking this question. I always enjoy reading the Address again (and I always forget how short it is).

Lincoln always had two reasons in mind for fighting the civil war. One was opposition to slavery. The other was preservation of the Union.

He saw the secession as a threat to democracy.

"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.

Now we are engaged in a great civil war, testing whether that nation, or any nation so conceived and so dedicated, can long endure."

Source: http://www.abrahamlincolnonline.org/lincoln/speeches/gettysburg.htm

In this great speech, that genius Abraham Lincoln managed to cover, in a few short words, both reasons for fighting the civil war: A democratic nation, 'conceived in liberty" cannot 'long endure' if the minority which is outvoted by the majority can just say, "Well, we didn't like that vote. We secede."

Democracy only works if a majority on a constitutional issue is binding. If it is not, democracy will not 'long endure' and anarchy, or some other form of government will replace it. So, "government of the people, by the people, for the people," would "perish from the earth."

The United States was "dedicated to the proposition that all men are created equal". Slavery is a denial of that proposition. Lincoln ran on a platform to ban the further spread of slavery into the territories. He didn't think (if elected) that he had the constitutional authority to ban slavery in states where it already existed, but he clearly thought it was incompatible with the essential tenets of the 'New Nation'.

It was, of course his election, bringing with it the promise of banning the spread of slavery, which caused the secession, threatening the concept of Democratic government.

Lincoln, quite eloquently expressed that the Union was fighting for two reasons, to oppose slavery where possible, and to preserve Constitutional Democracy as a form of government.

The war made the abolition of slavery possible (constitutionally) in the rebellious states (at least according to Lincoln's interpretation) and he took the opportunity and acted with the Emancipation declaration.

Victory in the war enforced the conclusion that secession due to a Constitutional majority vote was not allowable, thus preserving (in Lincoln's opinion) constitutional democracy as a form of government.

Thanks to both of you for these incredibly well thought-through and researched answers.