One reason why the supreme court and federal courts play such a big role in public policy is:

The Supreme Court plays a significant role in public policy because its rulings decide what is constitutional and unconstitutional in American society. In other words, the Supreme Court and the federal courts have the ability to interpret what laws mean.

A perfect example of this would be the United States vs. Butler case. This Supreme Court case from the 1930's states that the federal government cannot force farmers to stop making crops in order to decrease a national surplus. This shapes public policy forever, as this interpretation of a federal action as unconstitutional now limits the power of the executive branch.