和菓子の日 means the Wagashi Day. 和菓子, which pronounces as Wagashi, is a term for Japanese dessert. So June 16 is basically the Japanese Dessert Day.

The origin of Wagashi Day can be traced back to Emperor Ninmyo’s (仁明天皇) time in 848 AD, the year when he offered 16 different desserts at a ceremony on the 16th day of the 6th month in lunar calendar to pray for health. He also changed the name of the era to Kasho (嘉祥), a phrase from ancient Chinese poem that means auspicious. Since then the day had been called Kasho Day (嘉祥の日).

In 1979, the Japan Wagashi Association set the Wagashi Day to be June 16 (of the solar/Gregorian calendar, since after the Meiji Restoration Japan officially started using the Gregorian instead of the lunar calendar), for publicizing the charm of Japanese dessert.

On this day, Netizens tweets things about Wagashi on Twitter, causing the tag to trend.