、。
Since it was a matter of honor, he got angry.
Literal
Honor [to-に] relate problem [because-なので], he [topic-は] became-angry.
名誉にかかわる — built on the ~にかかわる ('be a matter of, affect') pattern, parallel to 命にかかわる ('be a matter of life'), but with 'honor' as the thing affected. 名誉 is a culturally weighty concept — 'honor' in the face-saving sense. ~なので is the polite/soft causal 'because.' 怒る as a past simple: 'he got angry' (not 'was angry').