Whole books have been written on this, and I find Frehling's two volume "Road to Disunion" books to be some of the best and most accessible evaluations of the American political landscape (as it evolved into secession) from 1776 to 1861. From the founding documents to secession, he covers a lot of ground to include the Dred Scott decision's ripple effect.
A shorter answer is as follows:
With this Supreme Court decision, abolitionists were provided grist for their public campaign against slavery because the institution of slavery, which some people in non-slave states put up with (so long as it was not legal in their state), was now viewed as being importable beyond its boundaries in the slave states. That meant that the Missouri Compromise might not stand up under the pressures from the Slave States.
That fear of the next Dred Scott decision shocked many in the North
who had been content to accept slavery as long as it was confined
within its then present borders. It also put the Northern Democrats,
such as Stephen A. Douglas, in a difficult position. The Northern wing
of the Democratic Party had supported the Kansas–Nebraska Act of 1854
under the banner of popular sovereignty. They argued that even if
Congress did not bar the expansion of slavery into those territories,
the residents of those territories could prohibit it by territorial
legislation. The Dred Scott decision squarely stated that they could
not exercise such prohibition, even though, strictly speaking, that
issue was not before the Court.
While some supporters of slavery treated the decision as a vindication
of their rights within the union, others treated it as merely a step
to spreading slavery throughout the nation, as the Republicans
claimed. Convinced that any restrictions on their right to own slaves
and to take them anywhere they chose were unlawful, they boasted that
the coming decade would see slave auctions on Boston Common. These
Southern radicals were ready to split the Democratic Party and — as
events showed — the nation on that principle.
Frehling's second volume does a fine job of addressing how the Secessionists in the South pursued their eventual ends, but that summary is sufficient for a short answer.
Why is this important?
The two influences on politics of ideas and their presentation in the press are often significant. When someone gets an idea and beats it to death in the media, they can whip up considerable popular and political support for their position.
- A classic case is Mothers Against Drunk Drivers campaign that began in
the 80's and has resulted in significant changes in the
DUI/DWI/Drinking and Driving laws all over the country. The power of
an idea and of getting a message out.
Another classic is the yellow journalism that got us "Remember the
Maine!" and helped grow support for the Spanish American War of 1898.
The political mileage that the abolitionists got out of this court decision was of value because they took advantage of it through the press and through public rhetoric. The ripple effect of the Supreme Court decision had a contribution (probably unintentional, from the PoV of the seven justices who supported it) to both the pro slavery/secession agenda, and to the abolitionist agenda.