The purposes are probably as many as there are protestors?

Wikipedia lists some of the organization participating. I don't think they try to keep it secret.

https://en.wikipedia.org/wiki/No_Kings_protests_(October_202...

The October 2025 protests follow the No Kings protests in June. Approximately 200 organizations have partnered for the October protests, including 50501 and Indivisible groups, as well as the American Civil Liberties Union (ACLU), the American Federation of Teachers, Common Defense, the Human Rights Campaign, Planned Parenthood Federation of America, Autistic Self Advocacy Network (ASAN), the League of Conservation Voters, MoveOn, Public Citizen, United We Dream, and Working Families Power.

Trump is arguing before the Supreme Court that these protests are "rebellion" and thereby authorise the use of force only intended to quell foreign invasions and civil wars.