While many Northerners were appalled by the South's "peculiar institution" and truly believed that abolition of slavery was an absolute necessity, it should be noted that the righteous arguments espoused by some Northerners were born more of convenience than of a deep-seated need to better the situation of Southern slaves. By the early to mid-1800's, slavery had largely died out in the North primarily because its practicality had vanished. Northern cities were more densely situated and had become far more industrialized than those in the South, which remained primarily agricultural. Thin, rocky soil had always made New England farming a less than lucrative venture, so industrialization replaced agriculture and slavery was more or less dying out in the Northeast as the cotton gin was making many Southern planters exponentially richer in the South. All the same, however, Lincoln issued the Emancipation Proclamation about halfway through the Civil War and shrewdly focused the attention of potential Southern allies like England on slavery by making it seem that the war was in fact being fought for freedom of the enslaved. Lincoln had once said that his primary objective was to save the Union, whether that meant freeing all the slaves, or none of that slaves. The Emancipation Proclamation "freed" slaves in the territories in rebellion--so in essence, not a single slave was freed--but it turned the focus of the war to liberty for slaves, which was exactly what the South didn't want.
When the South said they were fighting for liberty, it was in reference to their way of life (slavery) and states' rights, long a bone of contention between the two sections, going back as far as the Constitutional Convention. The South had always been in favor of nullification of federal law, and contended that they could not only nullify (ignore) federal laws they disagreed with, they could secede as needed from the Union, that this was a primary advantage to a "democracy" and "freedom". In fact, the Founding Fathers, as they were creating a nation that would "secure the blessings of liberty" at the Consitutional Convention had been forced to ignore the slavery issue for fear the document would not be ratified.
There are many different ways to define liberty and many different ways to define who is entitled to liberty. Thus, people can disagree on specific issues and yet still claim to both be supporting liberty.
As a modern example of this, think about the conflict over gay rights. Both sides would say that they are in favor of liberty. One side would say that liberty involves the right to be free from discrimination on the basis of sexual orientation. The other side would argue that gay people are not entitled to certain liberties (like marriage). They might also argue that liberty includes their own liberty to discriminate against people whose behaviors they disapprove of.
The issue of slavery was no different. Northern abolitionists believed that liberty should apply to all people. They felt that liberty could be defined, at least in part, as the right not to be owned by another person. Southern whites who supported slavery did not believe that all people were entitled to the same sorts of liberty. Just as we do not believe that children are entitled to liberty, these Southerners did not believe blacks were entitled to liberty. They also believed that liberty could be defined as the liberty to own slaves as property. Since that right had been implicitly supported by the Constitution, they could argue that they were simply looking to protect their liberty as defined in that document.