Relevant quote from Lincoln's First Inaugural Address:

It is seventy-two years since the first inauguration of a President under our National Constitution. During that period fifteen different and greatly distinguished citizens have in succession administered the executive branch of the Government. They have conducted it through many perils, and generally with great success. Yet, with all this scope of precedent, I now enter upon the same task for the brief constitutional term of four years under great and peculiar difficulty. A disruption of the Federal Union, heretofore only menaced, is now formidably attempted.

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. Continue to execute all the express provisions of our National Constitution, and the Union will endure forever, it being impossible to destroy it except by some action not provided for in the instrument itself.

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?

Descending from these general principles, we find the proposition that in legal contemplation the Union is perpetual confirmed by the history of the Union itself. The Union is much older than the Constitution. It was formed, in fact, by the Articles of Association in 1774. It was matured and continued by the Declaration of Independence in 1776. It was further matured, and the faith of all the then thirteen States expressly plighted and engaged that it should be perpetual, by the Articles of Confederation in 1778. And finally, in 1787, one of the declared objects for ordaining and establishing the Constitution was "to form a more perfect Union."

But if destruction of the Union by one or by a part only of the States be lawfully possible, the Union is less perfect than before the Constitution, having lost the vital element of perpetuity.

It follows from these views that no State upon its own mere motion can lawfully get out of the Union; that resolves and ordinances to that effect are legally void, and that acts of violence within any State or States against the authority of the United States are insurrectionary or revolutionary, according to circumstances.

I therefore consider that in view of the Constitution and the laws the Union is unbroken, and to the extent of my ability, I shall take care, as the Constitution itself expressly enjoins upon me, that the laws of the Union be faithfully executed in all the States. Doing this I deem to be only a simple duty on my part, and I shall perform it so far as practicable unless my rightful masters, the American people, shall withhold the requisite means or in some authoritative manner direct the contrary. I trust this will not be regarded as a menace, but only as the declared purpose of the Union that it will constitutionally defend and maintain itself.

From here it's clear that Lincoln is making an argument first for the perpetuity of the Union (meaning the illegality of secession) and that it is his official duty as President to preserve the Constitution that his oath swears him to uphold.

Next, stating that the South "explicitly didn't want to be part of [the Union]" is an oversimplification. Numerous scholars have demonstrated that the secessionists who spearheaded the push across the South to separate from the Union were a small and very radical political group. Their success was largely predicated on the extremely divisive events of the late 1850s (such as Harper's Ferry and the election of a Northern president despite essentially zero Southern support) which they could capitalize on to push their agenda.** As for the legality of secession itself, this was a highly contentious topic. While some figures (notably Southern firebrands and earlier figures such as John C. Calhoun) had argued for its legality according to the Constitution, others vehemently opposed it. Lincoln didn't suddenly make up the idea that secession was illegal in 1860-1, it was a highly contentious topic stretching back generations which had produced no clear answer. Hell, it's arguable how many people saw secession at first as anything more than an empty threat, because threats of disunion pervade much of the antebellum era (though, admittedly, declarations of secession were new, so this was clearly the farthest it had ever gone).***

Next, let's not forget that the South contained within it significant Unionist minorities, usually ranging from a quarter to a third of any given Confederate state.

But there's also a major factor on the other side of the Mason-Dixon line. As much as people like to discuss secession as an attempt by the South to preserve its constitutional vision, it must be remembered that the North had its own constitutional vision that (often but not exclusively, of course) aligned in many ways with what Lincoln states above. The Union was perpetual and the Constitution was sacred. Southern attempts to remove themselves from the Union amounted to a betrayal of the Constitution and the values it represented. Many Northerners were willing to fight to preserve it.

** A good source on this process is William Freehling's Road to Disunion series.

*** A great book on the history of disunion rhetoric is Elizabeth Varon's Disunion!.

Keep in mind that there was a secessionist movement in New England leading up to the War of 1812. If states could opt out, given enough time, it is conceivable that every state - or at least most - would sooner or later want to leave the Union because its interest diverged at a given moment. Allowing any state to leave the Union would mean that the nation would have no credibility in the long run and would probably have no future.

edit: remove "strong"

Politically and socially the Union and Confederacy were split, but economically the two depended on each other. The North was dominated by industry, had many factories developed and urban cities growing. Additionally they had modern railroads and the telegraph system in place. The South on the other hand, was mainly an agricultural economy. They depended on staple crops such as cotton and were mainly rural areas with large plantations. (One of the reasons why they depended on slaves, otherwise poverty in the region would deepen). Individually these regions would suffer, as they needed each other to develop economically. Even though the South had more lucrative trade during the Civil War, their economy was in shambles after the conflict and need the North's aid during the Reconstruction period of 1865-77. Have a look at John Keegan, James McPherson and Edward Ayers for more in depth information.

There are most probably more reasons, but I simply wanted to highlight the economic factor. (Note that I am not an expert on American history). Hope this helps a bit.