キスの日 means Kiss Day.

On May 23 1946, the movie はたちの青春 (Hatachi no Seishun, or Youth of the Twenty (Years Old)) was released, in which there was a kiss scene. It happened after WWII, and Japan was under the command of GHQ (General Headquarters of the SCAP, Supreme Commander of the Allied Powers, which was the US led force that was the de facto administration of Japan). There was this organization CIE (Civil Information Education Section) under the GHQ that managed many aspects of Japan, including movies. Both censorship and propaganda were implemented by CIE.

はたちの青春 was produced under this background as a propaganda film to a certain degree. It was said that the kiss scene was put in the movie by the order of CIE. As a result, the movie became a blockbuster, since oriental culture had always been conservative on such topics.

And thus the date May 23 was named Kiss Day (apparently not by any official means). Nowadays netizens in Japan tweet things related to kissing on Twitter, causing the tag to trend.