The basis of appeal for Homer Plessy was that the state law that allowed for segregated rail cars for black and white passengers violated the 14th amendment of the US Constitution. More specifically, Plessy argued that this violated the Equal protection clause of the 14th amendment.
In Justice Brown's opinion, he argues that the 14th amendment is not being violated in this scenario. This is because his interpretation of the 14th amendment is that this equal protection clause applies to the legal rights of an individual, not their social rights. In other words, Plessy is legally able to ride the railcar just like any other person. Therefor, the 14th amendment is not being violated.