My mother told me why my father was so angry at me.
Literal
Mother [topic-は] why father [subject-が] so me [to-に] angry-[ongoing-ている] [question-のか] taught-[benefactive-てくれた]
なぜ父がそんなに私に腹を立てているのか = embedded question: 'why my father is so angry at me.' 教えてくれた = 'taught/told me (as a favor).' のか marks the end of the embedded question clause.