Why did the south believe they could win the Civil War?
Remember at the beginning of the war, everyone thought it would be over very quickly. Many soldiers on both the North and South enlisted for 90 days or 1 year. The South thought the Northern states were not interested in going to war to fight to keep control of the Southern states. They felt that the Union soldiers were not dedicated as much as they were to defending their rights and homeland. The South quickly raised soldiers who were ready to fight. They had many officers and generals who were experienced soldiers & leaders. This was proven by the battles won by the Confederacy in the first year of the war.
Even after 3 years of fighting, the Confederacy was still seeking support from England. They felt that if England recognized them as a separate country, then that would legitimize their government and they would have support needed to continue to fight. This never happened.
Both sides in the conflict did not really understand the strength of feelings they each elicited on the other side. The North did not really believe the South will secede from the Union, since this had been threatened many times before but not actually carried through. The South, on the other hand, did not believe the North would go to war over the preservation of the Union since in every other respect the South was not actually threatening the Northern way of life. From their point of view all they wanted was to be left alone.
Interestingly, Lincoln was well aware of the weakness and intangibility of the North's cause and it is this which led him (among other reasons) to issue the Emancipation proclamation. Freedom is one of the few truly universal ideas for which people have been truly been willing to die.
The other point to remember was that the South saw itself as a far superior producer of soldiers than the North. This was not entirely inaccurate as the South did have the better officer cadre and its soldiers, raised in the wilder agrarian environment of the South , were more used to the rigors of military life. The fact that the North had the better industrial infrastructure necessary to support a War Economy was not immediately apparent and would only become apparent later on in the War.