The founders of our public school system meant by education exactly what Jefferson meant by education and exactly what thoughtful men had always meant by it.
网友理解正确。具体而言，第一个 meant 的宾语是两个并列的宾语从句：exactly what Jefferson meant by education and exactly what thoughtful men had always meant by it.
第二个和第三个meant 的宾语，都是各分句中的 what。两个 what 都是引导名词性关系分句。
