“He said that she killed people” vs. “He said that she kills people.
“That she kills people” was true when he (=the speaker reported, not the reporter) said it.
The truth of the report, “He said that she kills people” is impossible to state.
“He said that she kills people” sloppily (and ungrammatically) reproduces the words of the original speaker verbatim. Skip “that” and you can get away with it in speech. Add a comma after “said” and the quotation marks in writing: He said, “She kills people.”