As pointed out in the 1860 census, roughly 38.7% of the Southern population (the states that seceded) were slaves. 30.8% of families in the South owned slaves.

That, in and of itself, is huge. It meant that even if you didn't own slaves, you knew someone who owned slaves, almost undoubtedly. And it meant that you were probably related to someone who did, too.

That aside, it's often said (and this thread says it too) that people fought because they had a stronger connection to their states than to the federal government or Union as a whole. This is true, and I won't dispute that it may have contributed. However, slavery was also a huge motivation, even among those who didn't hold slaves. Why?

Many of those who didn't own slaves or belong to slaveholding families (which were a majority) still supported slavery because it defended their way of life, was something they aspired to (they wanted to become rich and own slaves), and wanted to maintain social order as they saw fit. They, despite forming a majority of the Southern electorate, never even came close to getting rid of slavery...they hardly wanted to.

Slavery was more than just something you committed to in owning a slave, it was a way of life. The ideas of patriarchy and the justifications for slavery rooted in social darwinism were ingrained into the Southern populace, which greatly differed from the culture of the north at the time. Fighting to preserve that way of life became crucial, not only for those who owned slaves, but for those who aspired to and who hoped to uphold the Southern way of life.

This would later become an issue yet again in terms of debates for upholding the Southern way of life when the issue of women's suffrage came up, as an example. Resistance to the changes of patriarchal order (not just in terms of gender, but in terms of white males at the top of the hierarchy) and to changes in the ways of life that had persisted thus far made Southerners feel it was imperative for them to fight.

There was also an economic issue to it, as well, that factored into this "way of life" issue.

Southerners, especially before the war began, argued that it would be impossible to abolish slavery, because the federal government couldn't compensate them enough and because it would lead to racial violence if African-Americans were free. They were focusing on the example of the British West Indies, where the British compensated slaveholders monetarily after freeing the slaves. They argued that if the federal government were to try to abolish slavery, they'd have to compensate the slaveholders to the tune of $900 million to make up for their losses (even if this was inflated, there would still have been a huge economic loss to the South as seen after emancipation), and the federal government was only raking in around $25 million per year in revenue. So the economic factor, and cultural/hierarchy factor, definitely factored in just as much as the loyalty to one's state with regards to why Southerners felt they had to fight. Still, there was yet another dimension to it.

Southerners, in arguing for the sake of maintaining the hierarchy, also made the argument that releasing slaves would be harmful to society on the whole. Even those not owning slaves feared that retribution, race violence, and the like would engulf the United States. It was for that reason that many of them were more than happy to support colonization; they figured they'd keep the slaves, and ship the rest of the black freedmen and freedwomen to colonies like Liberia. This idea eventually fell out of favor before the war, though...but that didn't stop the Southerners from fearing retribution if they lost control of their slaves, as well as race wars.

Main source:

Henkin, David. Becoming America. S.l.: Mcgraw-Hill, 2014. Print.

Before the Civil War, citizen's tended to feel more loyalty to their state than to the federal government. Robert E. Lee, for example, resigned his commission in the U.S. Army only when his home state of Virginia seceded. It's hard for us to understand how, but until the late 19th century, the U.S.A. was more of a confederation of independent states, as opposed to one nation. Soldiers served and fought for the south for the same reason that anyone fights for their home: national pride.

An article from the West Virginia Historical Society put it very well:

"In the South, 385,000 families owned slaves, out of a white population of 1,516,000 families.1 In the Army of Northern Virginia, for example, the majority of soldiers did not come from families that even had a direct personal stake in slavery. Therefore, "it was not the issue of slavery for which the average officer or enlisted man went to war." Actually, what really motivated them to enlist was their tremendous pride in their own land and what they and their fathers had achieved, "combined with a general dislike of Northerners stemming from most superficial knowledge of the real people who inhabited the northern states".

The article also references my earlier point about Lee when it talks of the motivations of senior Confederates:

Many high-ranking Confederates showed reasons for enlisting other than slavery. The examples consist of generals (or future generals). Robert E. Lee believed in neither slavery nor secession, but would fight for his old Virginia. Ambrose Powell Hill, better known as A.P. Hill, chose to fight for the defense of his state, Virginia, even thought he was deeply opposed to slavery. John C. Breckenridge, of Kentucky (a border state), a one-time Vice-President of the United States, sided with the Confederacy primarily for his home-state's self-defense from the North. The individual motivations are endless.

There were other motivations at play, but for the average Confederate soldier, the motivation was defending their home. (Remember, the South seceded and felt that they had the right to leave the Union. It was the North that initiated the war by refusing to sell back federal lands that were located in the south)

TLDR: Southerners identified as citizens of their home state, not of the USA. They fought for their home state for the same reasons that anyone fights to defend their nation.

Edit: Added TLDR

The Southern middling class - the yeoman farmers - were bound up with the plantation system. Besides the fact that many of them owned or rented slaves, they saw themselves as being in the same group as planters - free, land-owning white males. Throughout the mid-19th century, at least in South Carolina, greater and greater legal and political emphasis was given to the right of property and the owner's absolute authority over it, while at the same time restricting common rights (the ability to hunt on unfenced land, to cross another's property to reach your own). The idea of the head of household as the absolute master of his own property and all who lived upon it (wife, children, white servants, black slaves), developed. Thus, an assault on slavery, on the right of planters and prosperous farmers to keep their property, was an assault on their authority as well. This is to say nothing of white supremacy, the fear of slave revolts, and, among the landless or very poor whites, repugnance at the thought of competing with free black laborers. The very best book on the topic, devoted to why the yeomen fought, is Masters of Small Worlds by Stephanie McCurry.

Additionally, it should be noted that, while only 25% of southern households owned slaves, soldiers who volunteered for the Confederate Army were something like 40% more likely to be from these slave-owning households.

Here is a map (.pdf warning!) that shows the percentage of slaves (not including free blacks) in Southern states by county. As you can see, 32.2% of the population was enslaved, and two states (South Carolina and Mississippi) had more slaves than free men so it's easy to see that if slaves were freed and given voting rights the political landscape would be dramatically altered.