Freedom of religion. The Amish felt that requiring their children to attend school until 16 would be so disruptive of their way of life (and that their religion was their way of life) that the state requirement of mandatory schooling should not apply to them beyond the age of 14. I think the Supreme Court was very charmed by the Amish and wanted them to be able to continue their lifestyle and knew that if their children were required to attend school beyond that point, it would mean that Amish could no longer maintain the separation from the world that their religion required.
Do not rely on Wisconsin v. Yoder for precedent though. Other groups have tried and were rejected. It is an exception that essentially only applies to the Amish. Otherwise, the state has a sufficient interest in the welfare of children that it can force parents to school their children.
Anyway, now there is homeschooling. At the time of Wisconsin v. Yoder, homeschooling wasn't really legal, or else the Amish might have used that route. The Amish do have their own schools for grades 1-8. Most don't go beyond this because it is considered too worldly and they start working at that point.
During the 60s, there were several incidents where school authorities tried to round up Amish children and force them to go to school and where Amish parents were imprisoned for school attendance laws. It was very negative publicity for the states and school boards and so many areas reached some sort of accomodation with the Amish to respect their beliefs.
I hope this is helpful.