How are court decisions translated into actual policy?
Court decisions are most typically translated into policy by actions of the executive branch. Once a court has ruled on some matter of the law, it is up to the executive branch to enforce that decision.
Let us look at the issue of school desegregation. When the Supreme Court ruled that segregation was illegal, it was then up to local school districts (part of state executive branches) to implement and enforce the decision. Some school districts refused to implement the decision, with the most famous example being that of Little Rock, Arkansas. In that case, the US executive branch had to step in, sending troops to Little Rock to enforce the decision.
This is a very important thing to know about US politics because it limits the power of the judiciary. If the judiciary hands down a decision that is really unpopular, the executive branch can refuse to enforce it. If the people do not force the executive to do so, the decision will never be enforced.