I'll give more details below, but if you want a good summary, then seek out the book Lincoln and the Decision for War: The Northern Response to Secession by Russell A. McClintock. Specifically, the book's "Conclusion" section summarizes the reasons Lincoln and the North came to for engaging in war, though the whole book explains how and why these conclusions were reached. This paragraph, in particular, should answer your question about as succinctly as possible:

"Seeing four possible outcomes of the crisis—compromise, peaceable secession, the use of force, or a Southern retreat from disunion—[Lincoln] reasoned something like this: A) Northerners could not submit to a compromise, because 'buying' the right to take power after a constitutional election would not only encourage the rapacious appetites of the Slave Power but undermine the whole electoral system and destroy republican government. B) Northerners could not acquiesce in peaceable disunion, because secession was 'the essence of anarchy' and would destroy republican government. Therefore, C) either secessionists must cease their efforts to break up the Union, or Northerners would have no alternative but to preserve the republic by force."

If you want a more detailed answer, keep reading. The most often-cited reasons by the North are in bold.

The future existence of the United States was at stake. This was the primary motivation, which is why the U.S. armed forces during the war were referred to as "the Union". This was a reference to the "union" of states as laid out in the U.S. Constitution, since the U.S. was attempting to preserve the Constitution. The American Revolution had allowed the U.S. to be founded upon a free, "republican" (small "r") form of government, ruled by the popular will. Allowing some of the states to withdraw over the results of a perfectly legal and Constitutional election would doom the rest of the U.S. sooner or later. It would set a precedent that any other state could withdraw over some perceived political grievance, and there would soon be nothing left under the U.S. Constitution to preserve.

The U.S. Constitution was founded upon the principle of majority rule, mostly, with some built-in protections for the minority. If a minority can withdraw, through a "might makes right" proposition, then majority rule ceases to exist, and the very foundation of free government is at risk. A government founded upon a right to secession, as the Confederacy was trying to do, would lead to further secessions within itself. Free government would be replaced with despotism and anarchy. Neither the U.S. nor the Confederacy would long last in their present form, but would be replaced by something else, through force of arms, and would be something less than free. Lincoln said this very thing in his First Inaugural Address:

"If the minority will not acquiesce, the majority must, or the Government must cease. There is no other alternative, for continuing the Government is acquiescence on one side or the other. If a minority in such case will secede rather than acquiesce, they make a precedent which in turn will divide and ruin them, for a minority of their own will secede from them whenever a majority refuses to be controlled by such minority. For instance, why may not any portion of a new confederacy a year or two hence arbitrarily secede again, precisely as portions of the present Union now claim to secede from it? All who cherish disunion sentiments are now being educated to the exact temper of doing this.

"Is there such perfect identity of interests among the States to compose a new union as to produce harmony only and prevent renewed secession?

"Plainly the central idea of secession is the essence of anarchy. A majority held in restraint by constitutional checks and limitations, and always changing easily with deliberate changes of popular opinions and sentiments, is the only true sovereign of a free people. Whoever rejects it does of necessity fly to anarchy or to despotism. Unanimity is impossible. The rule of a minority, as a permanent arrangement, is wholly inadmissible; so that, rejecting the majority principle, anarchy or despotism in some form is all that is left."

This wasn't an unfounded fear. Not only was the South seceding, but early on, there was open discussion out West in California of whether to stay within the Union if secession should be accomplished, and instead form "a Pacific nationality". And nor was this anything new. The whole antebellum period had been marked by threats of disunion, and not only from the South (though mostly from the South, and nearly exclusively from the South from 1832 on).

Further, remember that four slave states (Maryland, Delaware, Kentucky, and Missouri) hadn't seceded yet. So negotiating a peace would put these states in a position of A) deciding which country to join, and B) if they stayed in the Union, they would continue to threaten secession ever after, and could hold it over the federal government as a coercive measure to get their way in political matters. There had actually been a book written by a Southern nationalist all the way back in 1836 that had predicted this exact situation. Nathaniel Beverley Tucker's work of speculative fiction The Partisan Leader predicted (in actuality, his way of advocating for) that the Deep South states would secede all at once, and then Virginia would join them later, threatening war if they weren't allowed to go in peace. And Tucker wasn't some crackpot - he had been a circuit court judge and a law professor at the College of William and Mary.

And even further than that, it's important to remember the international context of the time. The United States was considered an "experiment" at its founding, and there was still a very real concern that it could fail. The French Revolution hadn't happened long after the American Revolution, also aimed at founding a republic, only to be followed by the Napoleonic Wars and the establishment of the French Empire. The second French Republic had been established in 1848, but collapsed three years later in a coup d'etat, resulting in the second French Empire. That's where France stood in 1861, and that's what supporters of the Union feared would be their future - an unending series of conflicts in which monarchs and despots ruled, not free governments. In March 1861, right before the war broke out, a French diplomat was reported to have said to his American counterpart that the United States "was going to pieces", which was no surprise because:

"No Republic ever stood so long, and never will. Self-government is a Utopia, Sir; you must have a strong Government as the only condition of a long existence."

It's this very subject that Abraham Lincoln was talking about throughout his brief Gettysburg Address. It's short enough you should re-read the whole thing. When he's talking about the nation being "conceived in Liberty" and the Civil War is a test of "whether that nation, or any nation so conceived and dedicated, can long endure", he's talking about the future of free government, where the people themselves elect their leaders, and the majority vote is the government instituted, every four (and two) years. The Union is fighting so that "government of the people, by the people, for the people, shall not perish from the earth".

And added to all that, there was always the future prospect of international intrigue if the United States should dissolve. Britain and France may entangle the U.S. and the Confederacy in alliances that would see one or the other or both of them become possessions/colonies of European powers once again. Canada was still a colony of Great Britain at the time. France had intervened in Mexico's war of independence, and was threatening to invade them around the same time the U.S. Civil War started, which ended up commencing as a war between France and Mexico by the end of 1861. All the Caribbean colonies were still in the possession of Great Britain, France, Spain, or the Netherlands. There was a serious fear that disunion of the United States could re-capture the whole continent back under European control.

While the future of the nation was the main reason the United States engaged in the war, there were several other besides, though they mostly relate back to that primary motivation.

Any peace agreement would be temporary, and with war already inaugurated, now was the time to win it. This goes back to the reason above, that future wars would occur as the disunited states began to fight among themselves. But even if not, even if the U.S. and the Confederacy could somehow stay as two united countries, their differences were fundamental enough that they could not be resolved by just waving goodbye at each other. I've expanded on this topic in an earlier post, but to give the gist of it: secession is not the end of negotiations, any more the the UK's Brexit vote four years ago was the end. It was only the beginning. All the issues that caused disunion at the outset of the Civil War still had to be settled: how is the Western territory to be split up? How will the "Fugitive Slave" laws be enforced between two separate countries? Who keeps Washington, D.C.? How is the national debt to be split up? Does the South owe compensation to the rest of the U.S. for federal (tax payer-owned) property such as forts and arsenals they will be taking with them? How is navigation along shared waterways (the Ohio and Mississippi Rivers) handled, particularly as it comes to slavery?

cont'd...