1. on 是副词，表持续。比较：

stay at school = 呆在学校

stay on at school = 继续呆在学校

2. 例句如：

I stayed on an extra two years at school. 我继续在学校呆了两年。

I won’t go home yet; I’ll work on a bit. 我还不想回家；我要再继续工作一会儿。

We decided to play on even though it was snowing. 尽管下雪了，我们决定继续比赛。