The first and most important element of the case is that it invalidated the concept of Popular Sovereignty, whereby the citizens of each territory might have chosen for themselves whether slavery would be allowed. Chief Justice Taney's opinion declared popular sovereignty a violation of the Fifth Amendment. His decision removed any possibility of compromise in the territories.
Secondly, the decision took away from Congress the power to deal with the slavery issue. This had been a hot button issue all along, and it is problematic that Congress would have been able to do so; yet the decision meant that only a constitutional amendment could end slavery. Most constitutional scholars of the day, including Abraham Lincoln, believed that slavery was constitutionally protected where it existed. Taney's decision removed the possibility of Congress dealing with it in the territories.
Thirdly, his decision institutionalized slavery within the entire United States, even though many states had laws prohibiting slavery, and to my knowledge those laws were never changed. If one pursues Taney's decision to its logical conclusion, then slavery would be legal whether it existed or not. It became an iron clad right under the Constitution.
Slavery was such a hot button issue, by removing it from public debate except as a possible Constitutional Amendment, Taney virtually assured that the issue would be resolved only by bloodshed.
There are many causes to the Civil War. If we were to focus on the Dred Scott decision, I think that the primary reason would rest in the idea of an unbridgeable chasm between North and South. The abolitionist forces in the North could not reconcile their position with that of the Supreme Court's. Roger Taney's fateful declaration that suggested that slaves were not citizens and could not sue and that slavery could not be prohibited in the territories added to the tension between North and South in that it invalidated many of the Compromises reached in order to preserve the balance between so-called "slave" states and so-called "free" states. Southerners viewed the decision as a validation of their practices and a repudiation of abolitionist forces. This helped to increase the tension between North and South because the decision was so one-sided in its assertion, delivering even less chance of common ground. The decision incited already infuriated abolitionists in the North who became convinced that compromise on the slave question was going to be impossible.