"All rights reserved" is a copyright formality indicating that the copyright holder reserves, or holds for its own use, all the rights provided by copyright law. Originating in the Buenos Aires Convention of 1910, it is unclear if it has any legal effect in any jurisdiction. However, it is still used by many copyright holders.
The phrase 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.
Other copyright treaties did not require this formality. For example, in 1952 the Universal Copyright Convention (UCC) adopted the © symbol as an indicator of protection. (The symbol was introduced in the US by a 1954 amendment to the Copyright Act of 1909.) The Berne Convention rejected formalities altogether in Article 4 of the 1908 revision, so authors seeking to protect their works in countries that had signed on to the Berne Convention were also not required to use the "all rights reserved" formulation. However, because not all Buenos Aires signatories were members of Berne or the UCC, and in particular the United States did not join UCC until 1955, a publisher in a Buenos Aires signatory seeking to protect a work in the greatest number of countries between 1910 and 1952 would have used both the phrase "all rights reserved" and the copyright symbol.
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.