The chef was astounded by his unbelievable appetite.
Literal
Chef [topic-は] his [of-の] unbelievable [of-の] amount-of appetite [in-に] very was-surprised.
信じられない ('unbelievable') is the potential negative of 信じる ('to believe') — 'cannot be believed.' Used here as a modifier. 食欲 ('appetite'). に on 食欲に marks the cause/trigger of the surprise. A vivid character description.