Holden enjoys the snowball fight with the other students because it is a game of childish fun and he still has a strong childish streak in him. In fact the snowball fight shows that many of the other students are the same:
It was pretty childish, but everyone was enjoying themselves. (chapter 5)
Although these are high-school students, preparing to enter the adult world, they are still more than capable of breaking out into spontaneous fun, like children. This game represents some relief from the pressures of growing up. Also Christmas is very near, and the holiday mood is upon them.
Throughout the book, we see that Holden is really only able to relate to children: his kid sister Phoebe, the memories of his dead younger brother Allie, and various other children he meets in the course of the story. Perhaps most significant of these is the little boy whom he sees walking along a busy road, happy and carefree, singing the song which inspires the title of the book, and which, for Holden, comes to represent the whole glorious world of childhood. This is the world he is so reluctant to leave behind.