So, why in the sentence "The reason why he asked me that..." is a reducndancy or duplication of meaning.
However, sometimes you may come across the use of "reason followed by why". This is correct when the sentence sounds awkward if "why" is not there. For example, "There is no reason I should remain home." For the sake of smoothness, it would sound better by writing "There is no reason why I should remain home, or There is no reason for my remaining home."
(not a teacher)