The festival grounds where Yuka and Kouta spent Kouta's final day in Kamakura hold great significance to the plot of Elfen Lied. Most notably, it is the site of Lucy's mental breakdown and subsequent second massacre.
There is a high probability that the shrine where the festival takes place is the Tsurugaoka Hachimanguu Shrine, noted as being the most important Shinto shrine in the city. The festival is usually held for three days from August 7 to 9, making certain events take place during August 9th, the last day of the festival.
In the series
The festival first appears in Chapter 8 and Episode 3 during Yuka's flashback to her and Kouta's childhood. When they were children, Yuka made a bet with Kouta that he must do whatever she said if she succeeded in a papercraft cut-out game he excelled in at the festival.
During the extended flashback to Lucy's childhood, she wished to go to this festival with Kouta since it would be the last time they could spend together before he went home. Kouta apologized, saying he'd already promised to go with his cousin, but upon Lucy's inquiry as to the gender of his cousin, the boy couldn't decide whether to tell her the truth or not. He offered a white lie to spare her feelings, saying his cousin was male, and, pacified with that; the two children parted ways for the day with Lucy saying she would say goodbye to him at the station after the festival ended.
Unfortunately, Kouta's lie came to light as Lucy saw him comforting a distraught Yuka at the festival that night. Realizing he lied to her triggered a psychotic meltdown, during which Lucy gave in to the DNA Voice's instructions and slaughtered several people on the festival grounds. Her slaughter wouldn't be limited to the festival grounds, however. Not long after the festival came to an end from the assumed "bomb" threat people mistook her powers to be, Lucy would advance upon Kouta's family at the Gokurakuji Train Station and ultimately murder his father and sister on the train departing the station.