What advantage does this level of secrecy provide the justices?
I assume that you are asking about the secrecy that surrounds the Supreme Court justices when they deliberate about cases. If this is the case, the secrecy gives them an advantage because it helps to maintain their legitimacy in the eyes of the American people.
Each part of government needs to have some degree of legitimacy. That is, it needs to have people believe that it deserves to have power and authority. The Supreme Court is not elected, so it needs to get its legitimacy from some source other than popular consent. Political scientists believe that the Court’s legitimacy comes from the idea that it is apolitical and technocratic. People trust the Court because they believe that it will follow the law, not act in political ways. Secrecy allows the Court to maintain such an image. Because we are not allowed to know anything about how the justices decide their cases, we cannot become disillusioned with them. We can still believe that they are basing their decisions on high-minded factors, not on politics.
Thus, the secrecy gives the Court the advantage of being trusted by the people.