Clearly, in this question, you are asking about Scalia as an originalist and Breyer as an activist, respectively. The question would make no sense otherwise.
Originalists believe that the Constitution’s meaning depends on what the people who wrote the Constitution thought. They do not believe in the idea of having the Constitution evolve as Americans’ opinions evolve. If Americans want change, originalists believe, they can vote for it. In this case, if Americans no longer believe that capital punishment is moral, they should vote against it. The Court should not find that the Constitution says capital punishment is cruel and unusual when it clearly does not say so. In his concurrence, Scalia looks to the words of the Constitution and to his understanding of what the Framers meant when they wrote the 8th Amendment. He points out (on p. 2 of his opinion) that the people who wrote the Bill of Rights clearly thought capital punishment was acceptable because they said in the 5th Amendment that the government could not take someone’s life without the due process of law. By specifying when the government cannot take someone’s life, they imply that there are times when the government can take someone’s life. At the end of his concurrence, Scalia engages in originalism and judicial restraint. He says
The Framers of our Constitution disagreed bitterly on the matter (of capital punishment). For that reason, … they left it to the People to decide.
First, he is saying that we should be guided by what the Framers believed about capital punishment. Second, he is saying that if they did not clearly say whether capital punishment is acceptable, the Courts need to let the elected branches decide on its constitutionality.
Judicial activists believe that judges have the right to overrule the elected branches whenever they think it proper. They do not have to defer to the judgement of the elected branches. Your questions on this subject imply that your instructor also defines activism to mean that judges decide for themselves what the Constitution means instead of looking to the original intent of its writers. In his dissent, Breyer does not talk at all about what the Framers meant by “cruel and unusual punishment.” Instead, he talks about how things are in the US today. He mentions (on p. 2 of his dissent) “almost 40 years of studies, surveys, and experience” that, in his mind, show that the death penalty is flawed. He points out that most states in the US no longer use it. In mentioning these things, he is being an activist. He is saying that the Court should look at current conditions and decide whether the death penalty makes sense. He is saying the Court does not need to look to original intent and it does not need to defer to the elected branches. If, in the Court’s judgment, the death penalty is no longer in line with American values, the Court should declare it unconstitutional instead of waiting until Congress or the state legislatures make that decision. He says that various factors, including his own experience in 20 years on the Court lead him to believe that the death penalty “now” is unconstitutional. To an originalist, this would make no sense. If the death penalty was constitutional 20 years ago and the Constitution has not changed, the death penalty is still constitutional today. By saying capital punishment is “now” unconstitutional, Breyer is saying that the meaning of the Constitution changes over time. This is clearly an activist statement.
In these ways, we can see that Scalia is an originalist and Breyer an activist. Scalia relies on what the Constitution says and what its writers thought. Breyer relies on his own views and on things like studies that have nothing to do with what the Constitution says or what its writers thought.