Why do people use "he said" in direct speech but not in indirect speech? For example: He said, "I go to school." This one as "reported" indirect...

The use of "he said" with the exact words he speaks,  "I go to school" is to credit the actual words the speaker says so that the reader knows who said what.  In indirect speech such as he said that he was going to school, you learn the same thing, but do not hear the words the speaker actually spoke.  In effect, indirect speech is a rewording or paraphrasing of a direct quotation.  The tag of "he said" is to indicate that a direct quotation is coming which is why the very next thing is the quotation marks to show the actual words which he spoke. 

The confusion you feel is because the definition of indirect speech is that something is "reported," but in your direct speech example (He said, "I go to school."), "He" is "reporting" that he went to school. The solution lies in identifying who is "reporting." In direct speech, the quoted speaker or character may be reporting something. In indirect speech, the writer or narrator is doing the reporting.

In direct speech, a quoted speaker or character may report something, "I watched the sunset," but they may also not report something, "Did you see that sunset?!" The function of reporting is not part of the definition of direct speech: sometimes direct speech reports, sometimes it doesn't.

In indirect speech, the writer or narrator always reports something: they always report what some else said or thought or did or felt. As the writer, I might say something in indirect speech like this: I was walking and I heard that she said--now wait a minute, let me report this accurately for you--I heard that she said that it snowed heavily at Badger Pass and that it was just right for skiing. The writer or narrator always reports in indirect speech. The quoted speaker or character might report something or might not report something in direct speech.   

I realize that this is a difficult question for you, and I hope that my answer has clarified the difference for you.  Try writing one direct and one indirect quote of something you yourself say and see if that helps you understand.  For example, you might say, "I feel so frustrated with this idea!" she said as she threw her book down.  Or, she said that she was frustrated with trying to understand this idea.  Good luck!