In my opinion, the only reason the South thought they could win the war was hubris, plain and simple. This hubris was based on the need of the plantation owners to believe that their way of life could be preserved in face of the Industrial Revolution. This hubris could be fostered and preserved because the majority of Southerners lived in an information bubble. Most of them had never been out of their region and experienced the industrialized and urban areas of the North. Most only perceived the North as "a threat to their lifestyle."

Though the South did have some advantages, such as superior military leadership, and motivation for their cause, it seems rather foolish for them to have thought they could win in the face of the superior resources of the North. This is exactly what happens though, when you live in an "Information Bubble" built by a self-interested affluent class who feels its interests greatly threatened by the changing world.

I would argue that the Southern idea that they could win the war could be an additional chapter to the book by Barbara Tuchman, "The March of Folly: from Troy to Vietnam." This is where it really resides.

The South thought it could prevail simply because they (initially at least) thought their objective could be accomplished without war. The split in the Union didn't start with Fort Sumter. If you'll note from this Wikipedia article on Fort Sumter, South Carolina seceded from the Union in 1860. Fort Sumter was attacked in April 1861.

And that was the prevailing attitude up until Sumter was attacked. In the famous (and somewhat infamous) Cornerstone Speech, delivered by Alexander Stephens, who was serving at the Vice President of the Confederate States, he said:

I was remarking that we are passing through one of the greatest revolutions in the annals of the world. Seven States have within the last three months thrown off an old government and formed a new. This revolution has been signally marked, up to this time, by the fact of its having been accomplished without the loss of a single drop of blood.

The initial idea seems to have been for the South to peacefully secede, which is why they thought they would be successful. They didn't want to challenge the North to a fight, at least initially. Shots weren't fired for several months.

There was a very similar war a few years earlier that was almost certainly considered a precedent by the South, and that was the American War Of Independence.

In territorial terms, the two wars were very similar. One side declared independence, and the other side, to defeat independence, faced the task of invading and conquering the newly independent territory.

In 1776 the task turned out to be beyond the strength of the world's foremost military power. Imperial Britain lost.

In 1860, civil war presented fundamentally the same problem to a country - the North - that was in military terms virtually unarmed. To defeat the South, the North would have to enlist 10% of it's male population, raise and train huge armies, build and supply a massively expanded navy, establish war industries, tax its population, suffer casualties on a huge scale for years, and take back the South foot by bloodsoaked foot.

It was the South's misfortune that the North did exactly that.

Who can blame them for not predicting that the North would go to all that trouble?

The majority of the economic and political elite of the Southern States were aware of the existential threat to their economy and culture—chiefly as these were the direct result of malapportionment so as to continue latifundia slavery.

Existential threats are necessarily those which threaten the existence of an agent. The results of a lost war were identical with the results of not fighting. The elite of the South did not need to inquire into the likely chances of their success.

There are reasons for the average member of the elite to believe in the possibility of success based on the popular imagination surrounding very unlikely military causes such as the Dutch Republic, The American Republic and the French Republic and Empire. It does need to be remembered that politicians and even generals are poor analysts of the likelihood of martial success.

Finally the 19th century thought of elites were infected with thoughts of Elan and right making might.

There were few more pretty important South advantages in addition to already listed in previous answers factors:

1. Far superior military education. It was only one northern-located military school - West Point military academy, federal institution, which was accepting students from all states, including South. There were at least three state-sponsored southern military colleges, with exclusively (or at least predominantly) southern cadets: The Citadel and Arsenal Academy in SC, and Virginia Military Institute.

2. More extensive pre-war firearm experience for general population. Urban population, dealing with the firearms less than rural, constituted much bigger portion in Northern states than in Southern. Also, exclusively southern slave patrol system provided some level of training and experience for white southerners.

3. The Confederate States accounted for seventy percent of total US exports by dollar value. Confederate leaders believed that this would give the new nation a firm financial basis.

4. Slave labor allowed military enlistment for higher proportion of white population

However, Confederate decision to start the war was determined not only by advantages in the war, but also by disadvantages at peace.

When representatives from six seceding states assembled on 4 February 1861 in Montgomery, the wall map they were looking at was showing projected new country consisted from 15 states. Secession and Confederation leaders believed that all slave-holding states would join Confederacy. If this happened, North likely would not risk military actions to suppress secession, and rather consider negotiation - Northerner manufacturers needed Southern cotton. In this case, Confederacy did not need to go to war at all.

This did not happened: April 4th Virginia secession convention votes 80-45 against secession. 7-state slave-holding Confederacy leaders found himself with multiple problems:

White population Less than 3 million, versus Union more than 24 million;

Dangerously high black population proportion;

Lost US fugitive slave law protection;

High financial debt to Northern and British banks;

Weak and undeveloped manufacture;

Northern border with no geographical barriers, difficult to defend;

Insufficient merchant fleet and non-existing navy; no vessel-building facilities;

Lost cotton-production monopoly: Tennessee, Arkansas, and North Carolina produced 
15% of country cotton. If they stay in Union, US Congress could introduce cotton 
import tariff to protect their cotton production against Confederacy. Northern 
cotton market would be lost for Confederacy. Also, Union cotton-growing states 
would have home merchant fleet advantage for international cotton trade, and could 
increase production of cotton.

With Virginia convention 4 April vote, peaceful way to expand Confederacy beyond original 7 states was all but lost. The was also risky military way. April 10 virginian Roger Pryor, Confederate sympathizer, in his speech urged Southerners:

“if you wish Virginia to be with you, strike a blow!”.

However, time was running out. When Lincoln arrived to Washington prior inauguration, he discussed with V irginia statesman possibility of evacuating Fort Sumter in exchange to Virginia stay in Union: "A state for a fort is no bad business.". Now, after Virginia convention voted not to secede, representatives from this state were heading to Washington for negotiation with Lincoln. If Fort Sumter were evacuated by US government, there would be no plausible excuse to start the war. So, April 12 Confederacy did strike the blow at Fort Sumter. In a month Virginia joined Confederacy, followed b three more states. In one month, with no additional military action, Confederacy was twice as big and few times more powerful as before. Not bad, but as it turned out - not good enough.

The vast wealth of the Southern plantations was intrinsically tied up with their ability to generate a fabulous annual income, at low risk. Once the Union blockade was established by autumn 1861 those income streams shrank, the associated risk rose, and the value of those plantations, dropped to a tiny fraction of the ante bellum value.

For a perpetuity (perpetual annuity), a reasonable ante bellum approximation,

PV = D / r

where: PV is the Present Value;
       D is the dividend per period; and
       r is the discount rate (assessment of risk)

With D dropping by 80+% and r increasing by a factor of 2 or 3, those fabulous plantations were suddenly worth less than 10% of their 1860 value; and even at that price there were no buyers available. Not only was direct income of the CSA hit by the blockade, but its fabulous pre-war wealth and the associated borrowing capability instantly vanished. In just a span of months the CSA went from being fabulously wealthy to destitute and bankrupt.

The expectation also existed that Britain and France would intervene to protect their growing textile industries. However with greatly shrunken supply, those mills were able to maintain profitably due to greatly increased prices, as well (ironically) for the immediately increased demand for cotton military uniforms required by both sides.

Finally, the South was unaware of the repugnance with which its slave-holding economy was regarded in Europe. This ensured that Britain and France would only enter the war to jump on an already victorious bandwagon, and not to decide the fray.

The only way in which this triple whammy could possibly have been avoided would have been to enter the war with an already extant navy capable of keeping at least a few of he South's major ports open against a blockade. In practical terms this likely meant a need for several years of pre-war independence. This was never going to happen under a Republican president, so it would have been necessary to secede under Buchanan and immediately begin a naval construction program, while he remained President.

As regards the first point of the paragraph above, Article I, Section 10, of the U.S. Constitution states in part:

No state shall, without the consent of Congress, lay any duty of tonnage, keep troops, or ships of war in time of peace, ....

Only a very friendly administration in Washington would have chosen to overlook the maintaining of independent "States' navies" so expressly forbidden in the Constitution.