What part does the Supreme Court play in interpreting the Constitution?

Basically any way their own personal experience and beliefs influence them to. Since they were selected, a lot of people trust their judgment. Notice the root of the Word judgment?

Another view: The terms"without respect to persons" and "impartially discharge" make it clear that judges are specifically not to make decisions on their own experiences or beliefs.

According to Title 28, Chapter I, Part 453 of the United States Code, each Supreme Court Justice takes the following oath: "I, [NAME], do solemnly swear (or affirm) that I will administer justice without respect to persons, and do equal right to the poor and to the rich, and that I will faithfully and impartially discharge and perform all the duties incumbent upon me as [TITLE] under the Constitution and laws of the United States. So help me God."