Why was the Plessy v. Ferguson case important to the era of segregation?

The Plessy V. Ferguson case in 1896 started segregation as a legal entity in the United States.  In 1892, Homer Plessy, a man who was one eighth black, violated Louisiaina's Separate Car Act which required whites and blacks use separate rail cars.  The penalty for violating this act was $25 or twenty days in jail.  Plessy's argument was that this law violated the Thirteenth and Fourteenth Amendments to the U.S. Constitution, the first of which prohibited slavery and the second one that is credited with making the former slaves American citizens. The majority opinion of the Supreme Court ruled that "the Fourteenth Amendment gave all citizens equal status before the law," but it said nothing about giving people separate access to public utilities.  This created the "separate but equal" amenities that would be in much of the South until the early 1960s.   There would be separate lunch counters, schools, water fountains, and bathrooms, and often these would not be of the same quality that whites got to use.