The Chief Justice and Associate Justices are appointed to the Supreme Court of the United States (SCOTUS) by the President with the advice and consent of the Senate. Therefore, the process is political because it is a lifetime appointment by a President who selects candidates based upon the nominee's judicial philosophy; does it align with that of the President?
The case of Marbury v. Madison, 5 U.S. 137 (1803) established the principal of judicial review - the SCOTUS decides the Constitutionality of legislation and actions of the Executive and Legislative branches of government. This has a significant impact, not only upon jurisprudence, but politically as well. For example, in 1973, the SCOTUS decided the case of Roe v. Wade, 410 U.S. 113 (1973) that protected a pregnant woman's right to have an abortion. At the time, the majority of of the Court were Democratic appointees, leaned to the left politically/philosophically and tended towards judicial activism - those who influence public policy and legislation through their rulings. Successive Republican presidents, with the opportunity to make appointments to the SCOTUS nominated individuals who tended to be strict constructionists - interpreting legislation and the Constitution only as written within the four corners of the document. Since the court now leans to the right, many states have enacted legislation restricting Roe v. Wade in the expectation that the SCOTUS will reverse or restrict Roe v. Wade should the new laws be challenged in the courts.
The opinions and decisions of the Federal courts and the SCOTUS reflect the times and changing attitudes and politics of American society. Although the Constitution was written over 220 years ago, the interpretations of the Constitution by the courts makes it dynamic and ever changing with the times. If the courts did not change, we would still live with the effects of the Dred Scott case (Dred Scott v. Sandford, 60 U.S. 393 (1857), where the Court decided the Constitution did not consider African-Americans to be citizens. Like society, the courts change to reflect the times and that is, by its nature, a political process.
This answer is by no means exhaustive. If you are interested, I suggest you take a course on Constitutional Law. It is a fascinating look at how American society develops by interpretation of our laws by the courts.