This case did not change the way that the Constitution or the amendments in general are interpreted. Instead, what it did was to clarify the meaning of one specific clause of the First Amendment.
Epperson v. Arkansas was a case about the teaching of evolution in public schools in Arkansas. The state had a law that banned the teaching of that theory. Susan Epperson was a teacher in Arkansas who wanted to be able to teach evolution. When she sued, the Court ruled in her favor, holding that Arkansas could not ban the teaching of evolution.
This decision changed the way that we interpret the Establishment Clause of the First Amendment. That clause says that the government cannot establish an official religion. In this case, the Court held that this means that the government cannot favor one religion or one religious group. It found that the only point of the law was to protect certain Christians from being exposed to ideas that they did not like. Therefore, it held, the law was an example of the government showing an unconstitutional degree of favoritism to one religion.