Final answer:
Interest groups influence the courts by initiating lawsuits, lobbying for judicial appointments, and filing amicus curiae briefs to present legal arguments to the courts, such as in the influential case of Obergefell v. Hodges.
Step-by-step explanation:
The influence of interest groups through the courts occurs through initiating lawsuits and lobbying for certain judges to be appointed to the bench. Interest groups engage with the judicial branch primarily by litigation, which involves making a case subject to a lawsuit, and by attempting to influence judicial appointments made by the executive branch and confirmed by the Senate.
Furthermore, these groups may file amicus curiae briefs, also known as ‘friend of the court’ briefs, to present legal argumentation to persuade the court’s decision-making in particular cases. A notable example of such influence was in Obergefell v. Hodges, where numerous interest groups filed amicus briefs that were instrumental in the Supreme Court’s ruling in favor of legalizing same-sex marriage nationwide.