The African-American community in Montgomery was able to respond quickly to Rosa Parks arrest because they were already planning on a way to protest against bus segregation. The NAACP and Martin Luther King Jr were already planning on having a member of their group challenge this law. Rosa Parks, at the time of her arrest, was a secretary for this group for the Montgomery chapter.
Rosa Parks was a well known community member and an advocate for civil rights already. Along with the fact that the NAACP had been planning a protest against bus segregation explains why it was so easy to garner support for the cause.