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.