Translation:She wants him to tell her the truth.
This is a complex sentence consisting of two clauses chained with "que". In the main clause "Ella quiere", "ella" is the subject. In the subordinate clause "él le diga la verdad", "él" is the subject, "la verdad" is the object and "le" (which from the context means "ella") is the indirect object. To whom does she want him to tell the truth? To her.
I absolutely agree with you and in a normal conversation I would say it like that as well. However at DL I try to keep it more on the literal side as long as it does not sound retarded or nobody would ever say that. So far that went decently well for me, but here it is gramatically incorrect in my opinion as you would have to say "he tells her".
"She wants that he tell her the truth" is accepted because it employs the English subjunctive tense, so 'tell' is correct. The subjunctive tense is seldom used in English today but in the somewhat archaic sentence structure in use in this example the English subjunctive tense is used.