Translation:He promised several people that he would teach them to read and write.
It's just another way of saying the same thing.
Czech prefers clauses, that's why "...že je naučí" is a lot more natural than "...naučit je". English, on the other hand, has no problem with "...to teach them" without the "...that he would teach them".
I wouldn't punish people who choose the shorter version in English, there is no difference in meaning.