Copyright law in most countries no longer requires such notices, but the phrase persists.
[1] It was required to add the phrase as a written notice that all rights granted under existing copyright law (such as the right to publish a work within a specific area) were retained by the copyright holder and that legal action might be taken against infringement.
Since copyright law is neither straightforward nor widely understood in its details – nor is the Buenos Aires Convention's previous requirement, and the current deprecation of the phrase, common or lay knowledge, it continues to hold popular currency and serve as a notionally-useful convention widely used by artists, writers, directors, photographers, designers, and other content-creators with the intention that it should function as both sign and warning that the content alongside which it appears cannot be copied freely.
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.
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.