It wasn't a good ruling at the time. I'm going to try and go through all the legal issues in the case and explain the issue, the ruling, the reasoning, and the problems with that reasoning:

The Facts

Dred Scott was a slave owned by Dr. John Emerson. Emerson took Scott to both Illinois (a free state) and Wisconsin (a free territory) before returning him to Missouri. The traditional explanation for slavery under the Common Law (based on the thinking of people like William Blackstone) was that people were naturally free and slavery only existed because a law authorized slavery. However, if a slave stepped onto the soil of a land that did not authorize slavery, he would become free. In the United States, if he escaped from slavery, he would not be free and would have to be returned because of the Fugitive Slave clause of Article IV of the Constitution. Since Scott had not escaped but was taken by his master to free soil, he argued that he should be free of slavery.

Scott sued for his freedom in Missouri court where he won in lower court but, ultimately, the Missouri Supreme Court reversed. This isn't technically part of your question, but I don't think the decision was even sound when Judged under Missouri law. Prior cases had recognized Scott's exact point. The Court's reasoning was that those old precedents didn't apply now because the actions of abolitionists have changed the status quo and are now threatening to undermine Missouri's government.

At this point, Scott was owned by John Sanford. More importantly, Sanford lived in New York. Since Scott was in Missouri, they were in different states. Under Article III of the Constitution, Federal courts could hear "Controversies ... between Citizens of different States."

Diversity of Citizenship Question

This is the first holding of Dred Scott. The Supreme Court ruled that Scott wasn't a "citizen" of Missouri because he was a black man whose ancestors came as slaves and, therefore, there was no diversity of citizenship that gave the Federal Courts jurisdiction. I'll get into the reasoning in a second, but it's worth pointing out that this should have completely decided the case. Once the Court made the ruling that it didn't have the power to hear the case, it should not have ruled on all the other issues presented. That being said, this has not been consistently applied and would not be the first or last case vulnerable to this criticism.

But was it right on the merits? Well, there's nothing to directly address the issue in previous Supreme Court cases. Justice Taney more or less cites no cases. Instead, he argues based on the Constitution and the Declaration of Independence. I think his argument is nearly nonsensical, but I'm certainly editorializing. Essentially, the Constitution described the "people of the United States," which Taney says is equivalent to citizen of the United States. He then says no state can make someone a citizen of the United States. If they're not a citizen of the United States, they can't be a citizen of a state without their consent. In other words, Dred Scott is not a citizen of the United States, Missouri says he's not a citizen, so he's not a citizen of Missouri. Then he said the history of the United States demonstrated that Africans imported to the United States were viewed as inferior and, therefore, not a part of the class of US citizens. Additionally, he presumes that slave holding states would not have ratified the Constitution if they believed that freedmen would be citizens.

Two Justices disagreed on this point. There are some technical arguments: basically, no one argued that he wasn't a citizen when they were supposed to. I'm not qualified to comment on this issue. I think a Court today would not find the issue waived, but it's possible things were different then. The more substantive arguments are pretty persuasive. Justice Curtis argues the history: there were people freed from slavery who were viewed as citizens at the time of establisment of our country. The idea that the founders didn't think freed slaves could be citizens is demonstrably not universal. Justice McLean argues something else - citizenship of a state for purposes of suing generally has nothing to do with whether they're a citizen of the United States. In fact, the discussion he has is pretty much the modern definition. If you have your domicile in a state, that's the state you are a citizen of for purposes of the question of whether you can sue someone from another state in Federal Court.

The Missouri Compromise

Chief Justice Taney didn't stop there, though. The next step was to declare the Missouri Compromise unconstitutional. Essentially, it turned on what power the Federal government had to regulate the territories of the United States. One answer is relatively straight-forward. Congress has the power to "make all needful rules and regulations respecting the territory or other property belonging to the United States."
The Court concluded that this language only concerned territory owned at the time of the treaty with Great Britain and tangible, moveable property. The reasoning is basically that, if the Constitution was giving power to create territories with rules, it would do so with more clarity. This argument is weak. The clause of the Constitution is part of the same clause that talks about admitting new states into the Union and doesn't at all discuss anything in particular about Great Britain. The dissent quotes Chief Justice John Marshall from Atlantic Insurance Company v. Canter who explictly ties the "needful rules and regulations" language to the territorial government of Florida (which was not acquired in the treaty with Britain).

Taney's reasoning doesn't flow from the language of the US Constitution. Instead, it flows from the mouth of southern theories of the Union that could be used to justify nulification and secession. Essentially, the United States held the property in trust for the states who were the ones that had joined together and ceded power to form the United States. Given that, Congress is limited in its powers to infringe on the rights of citizens, including the right to own property. This post is getting quite long, but I'll note that the Fifth Amendment doesn't prevent Congress from depriving of property, it prevents depriving property without Due Process of Law. The dissent cites a number of laws where the Constitutionality of banning slavery was not questioned including parallels to state provisions. One of the obvious examples if the banning of the Atlantic slave trade (something that was relatively uncontroversial at this point). Long story short on this point, John Marshall's opinion seems to address what provision of the Constitution gives Congress the power to regulate territories, but this certainly wasn't without controversy and secessionists thought they had legitimate arguments if you ignore precedent.

But What About Illinois?

So the Court's ruling has said that Scott's travel to Wisconsin didn't cause him to be free. But why do we even have to worry about territories, Scott went to Illinois - a free state! This is interesting because, if the case had been decided today, the answer would be clear and actually in Taney's favor (13th Amendment aside). Today, we have something called the Erie Doctrine, which says, in diversity lawsuits, you have to apply the law of the state in question. In other words, even though Missouri had previously held numerous times that entering free territory would have freed Scott, their ruling this time that it didn't free Scott is controlling. However, at the time, the controlling law was a precedent called Swift v. Tyson, which said that the Federal courts are to apply Federal Common Law to decide issues rather than state law. This means they are supposed to decide what happens when a slave goes to a free state. That being said, it's hard to entirely fault Taney here because there is a case called Strader v. Graham that lends support to the idea that the Supreme Court can't overturn the state's determination (as dubious as that determination was), so I won't say this decision was clearly wrong.