Before addressing the question of why the supporters of the Confederacy believed that their government's forces would win a quick victory over the forces of the United States, it should be noted that, regardless of what the people of the North thought, the United States itself was in no position to achieve a quick victory over the rebellion.
The United States certainly had access to far greater resources than those contained within the Confederacy, including far greater manpower, more industry, a more extensive infrastructure, and more wealth. But all of these resources would take time to harness to a war effort. The Regular Army of the United States at the outbreak of the war consisted of only 16,000 officers and soldiers (although few of the soldiers deserted to serve in the ranks of the Confederacy, most of those officers who were from the South chose to resign their commissions in the U.S. Army and seek a commission in the Confederate Army). As such, both sides would fight the war with field armies largely composed of volunteers with no prior military experience.
Now, for your question. The Confederacy entered the war (indeed, provoked the war by shelling Fort Sumter) with the belief that they most certainly would defeat the forces of the United States. They believed this for two reasons.
First, the leaders of the Confederacy believed that they held an ace up their sleeve called "King Cotton.' The leaders of the Confederacy knew their history and knew that, in order to win their independence, it had been critical for the thirteen Colonies to obtain the aid of a powerful European Ally. They believed that Great Britain and France had become dependent on shipments of cotton from the Slave States and, therefore, if this supply was interrupted, it would produce social (massive unemployment of mill workers) and economic turmoil in these countries. Thus, they believed that, in order to avoid the consequences of a lack of raw cotton, Great Britain and France would intervene in the war on the side of the Confederacy in order to restore their supply of cotton.
The declared establishment of a U.S. Navy blockade of the Confederate coast might have eventually produced the circumstance that the Confederate leaders felt certain would result in intervention by Britain and France. However, because the U.S. Navy lacked an adequate number of warships with which to make the declared blockade into an actual blockade (something it would take some time to remedy), the leaders of the Confederacy introduced a self-imposed embargo on the export of cotton to Europe to produce the required effects in Europe.
But there was a problem. In the years leading up to the outbreak of the war, the Slave States had sent so much cotton to Great Britain and France that these nations had roughly two years worth of cotton stored in their warehouses. It would be a long while before Great Britain and France faced the threat of social and economic turmoil due to a scarcity of cotton. Nor did the mill owners wait for this eventuality. Britain found that perfectly good cotton could be grown in Egypt and their colony of India. Moreover, what the leaders of the Confederacy failed to consider was that the volume of trade between Northern States and Europe was also of vital importance to Great Britain and France. The first two year of the war, for instance, coincided with a period of crop failures in Europe, so that Britain and France were dependent upon grain shipments from the United States. Lastly, Great Britain, without whom France declined to become involved in the U.S. Civil War, had become a leading force in the opposition to slavery, having abolished slavery in her colonies and having dispatched warships to forcibly stop the trans-Atlantic slave trade. This significantly reduced the likelihood of Great Britain coming to the aid of the Confederacy.
Second, Southerners in general had convinced themselves that they were culturally / ethnically far superior to the people of the Free States. Northerners, in the eyes (and publications) of Southerners, were a low species of money grubbing scalawags who lacked all sense of honor. In fact, there was really nothing that one could accuse Northerners of being in the company of Southerners in the decades leading up to the war that wouldn't be met with their approval. As one saying at the beginning of the war had it, 'One southern soldier is worth ten Yankee hirelings.'
Needless to say, one of the cardinal sins that leaders can commit during a war is to underestimate the enemy, and this was a cardinal sin that the supporters of the Confederacy committed in spades. On the days leading up to the First Battle of Manassas / Bull Run (21 July, 1861), the Confederate officers and soldiers harbored no doubt whatsoever that once they opened fire on the detestable Northerners, they would immediately take to their heels like the cowards they were. Of course the 'soldiers' (civilians in uniform) of the U.S. field army under Gen. Irvin McDowell were no less naive and no less confident of their forthcoming triumph (though a Confederate victory, the first major battle of the war was actually a very close run affair during which the Confederates were on the verge of defeat on more than one occasion).