The doctrine of states' rights held that the states had the right to make their own laws and regulations, even if those laws conflicted with federal law. This doctrine was often used by Southern states to justify their refusal to desegregate schools, public transportation, and other public facilities.
The words "interposition" and "nullification" refer to two specific ways that states' rights were used to justify segregation. Interposition held that states had the right to interpose themselves between the federal government and its citizens, and to nullify federal laws that they deemed to be unconstitutional. Nullification held that states had the right to simply declare federal laws to be null and void within their borders.
King's use of the phrase "dripping with the words of interposition and nullification" was intended to highlight the way that these doctrines were being used to maintain a system of racial segregation and discrimination. He argued that these doctrines were not based on any sound legal principles, and that they were simply being used as a way to prevent African Americans from exercising their constitutional rights.