The Dred Scott case supported the right to own slaves. So why is the case a reason that the South seceded from the Union?
The decision in this case was not in itself a reason that the South seceded. The decision was, as you say, favorable to the South. As the link below tells us, "Southerners were for the most part gleeful" with respect to the decision.
The case did, howeer, help lead to secession because it made it so that the two sides could no longer compromise. Part of the decision said that Congress could not make any laws about slavery in the territories. This meant that laws like the Missouri Compromise were unconstitutional. It also meant that no more such compromises could be made and there might have to be conflict (as in "Bleeding Kansas") over the issue of slavery in each and every territory.
So, the part of the decision that was pro-slavery did not cause secession. It was the part that banned Congress from making laws on slavery in the territories that was important. It made conflict between the North and South much more likely and took away the best way (compromises) for resolving tensions. This made secession more likely.
Stephen Douglas, in his Freeport Doctrine, explained (however awkwardly) how popular sovereignty was reconcilable with the Dred Scott decision. Territories, he claimed, could get rid of slavery by refusing to enact laws supporting it. This position alienated the southern states. The decision also mobilized support for the Republican Party, which was seen by many as an alternative to a "slave power" that had an influence over government that was disproportionate to their numbers. In 1860, the Democratic Party split as southern radicals refused to support Douglas due to his continuing support for popular sovereignty. Once this happened, Lincoln, the Republican nominee, won the election. Many in the Deep South feared he would use the powers of patronage to appoint anti-slavery officials who would disrupt slavery in the South as well as enacting measures to weaken the institution in the territories. They left the Union out of frustration that they could no longer use the power of the federal government to protect their interests.
The decision in the Supreme Court Case Dred Scott v. Sanford 1857 added to the disintegration of compromise between the north and south. The court ruled that slaves were property, basing their decision on the 5th Amendment. The 5th Amendment prohibits the Congress from depriving citizens of life, liberty, or property without the due process of law. This decision by the court made the Missouri Compromise as well as any other attempt to keep slavery from expanding unconstitutional. The Dred Scott decision made slavery fair game in any territory. As a result of this decision the south, especially South Carolina justified their secession because Lincoln wanted to stop the spread of slavery into the west, potentially challenging the Dred Scott decision.