Abraham Lincoln explained his reasoning on this point at length in his first inaugural address. To quote some of the central points of his argument:

I hold that in contemplation of universal law and of the Constitution the Union of these States is perpetual. Perpetuity is implied, if not expressed, in the fundamental law of all national governments. It is safe to assert that no government proper ever had a provision in its organic law for its own termination...

...Again: If the United States be not a government proper, but an association of States in the nature of contract merely, can it, as a contract, be peaceably unmade by less than all the parties who made it? One party to a contract may violate it—break it, so to speak—but does it not require all to lawfully rescind it?...

...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 remained the official view of Lincoln's administration throughout the war. Slavery was abolished, in the words of the Emancipation Proclamation, "as a fit and necessary war measure for suppressing said rebellion." The public did move beyond this position to view the war as being ultimately to end slavery (as framed in Bancroft's 1866 memorial address before Congress), but that was never officially recognized by the government and did not take place until well into the war.

I believe it's Adam Goodheart who argues in 1861: A Social History Of The Civil War, that we should remember how fresh and unproven the American experiment in self-government was. If the Southern states were allowed to leave the Union, what would stop New York or Massachusetts from leaving a few years hence over some perceived slight or economic imbalance? The idea of Union was not as remote and abstract an idea to fight for in the 1860s as it seems to us today, after a century of thinking of the US as a single nation tightly bound together economically.

We know how the experiment turned out—with a strong continental nation—but at the time there was no such certainty. If states could secede, they might be at danger of being absorbed by European powers, possibly eventually bringing the endless wars of Europe to North America.