First, we must understand what the Dred Scott case was truly about.
The Dred Scott case (Dred Scott v. Sandford, 1856 - 1857) questioned the status of slavery, and whether or not individual rights (or non-rights) was placed over states rights or not. In the case of Dred Scott, his owner moved from Missouri (which allowed for slavery), to Illinois (a free state). Dred Scott, on arriving, sued for his freedom, stating that living in a free state allowed him to become a free man.
The Court decision appealed Dred Scott's case in multiple reasons, including the fact that since Dred Scott is not a citizen, he could not use the American judicial system. Also, slaves were ruled to be as property, not human, which protected slavery under the 5th Amendment, which states that "No person shall be.. compelled in any criminal case to be a witness against himself, nor be deprived of... property...".
Why is this important?
The case increased tensions between North and South states, as the North saw it as a infringement upon their territorial laws. After all, the state laws in the North typically banned slavery, but, and when, as in the case of Dred Scott, Southerners who moved to free states can void the state laws, it infringes upon the values of the people who voted the laws in place. Accordingly, slave catchers also started to roam the Northern states in hopes of catching runaways, which led to the North to be not as safe for slaves as before. This is one of the reasons why the Underground Railroad, which used to lead to Northern states, was then extended to British-controlled Canada, where slavery is outlawed.