"All rights reserved" is a phrase that originated in copyright law as a formal requirement for copyright notice. It indicates that the copyright holder reserves, or holds for their own use, all the rights provided by copyright law, such as distribution, performance, and creation of derivative works; that is, they have not waived any such right. Copyright law in most countries no longer requires such notices, but the phrase persists. The original understanding of the phrase as relating specifically to copyright may have been supplanted by common usage of the phrase to refer to any legal right, although it is probably understood to refer at least to copyright.
The phrase appears to have originated as a result of the Buenos Aires Convention of 1910. Article 3 of the Convention granted copyright in all signatory countries to a work registered in any signatory country, as long as a statement "that indicates the reservation of the property right" (emphasis added) appeared in the work. The phrase "all rights reserved" was not specified in the text, but met this requirement.
The requirement to add the "all rights reserved" notice became essentially obsolete on August 23, 2000 when Nicaragua became the final member of the Buenos Aires Convention to also become a signatory to the Berne Convention. As of that date, every country that was a member of the Buenos Aires Convention (which is the only copyright treaty requiring this notice to be used) was also a member of Berne, which requires protection be granted without any formality of notice of copyright.
The phrase continues to hold popular currency and serves as a handy convention widely used by artists, writers, and content creators to prevent ambiguity and clearly spell out the warning that their content cannot be copied freely.