Japanese culture has a fondness for homonyms (a word that sounds the same or is spelled the same as another word but has a different meaning), which is how we get to the Japanese concept of "meat day". The numbers "2", pronounced Ni as in "need", and "9", pronounced Kyū as in "cuckoo clock"; combine these two singular numbers together (like saying two nine instead of twenty nine) and the result sounds like the Japanese word for meat (Niku). This is why the 29th day of each month came to be considered by some as meat day:
Leap years create a special situation in Japanese culture, where February 29th has come to be known as Japan's national meat day, pronounced "uruo niku no hi":
Here's a more concise description of Japan's national meat day from The Japan Society:
"In Japan there is a curious word play regarding the 29th day of the month in Japanese, put into just its number components it creates the word ni-kyuu [二九] which is the same as niku [肉] which means meat. So, quite often many people can be found eating meat on this day in Japan. Delving into the mystery of 'niku' day turns up a number of interesting facts about Japanese culture, and of course meat. For example, the 9th of February is also a day of meat [肉の日] due to it being the second month and the ninth day, ni-kyuu [二九], and the 29th of April is a day for eating lamb and mutton [羊肉の日]. More importantly, the 29th of February only occurs every four years so it is known as uruo niku no hi [閏肉の日], a leap day of meat and therefore a special day."