。
She despises me on the grounds that I have no sense of humor.
Literal
She [topic-は] me [object-を] humor [of-の] sense [subject-が] not-exist [the reason being-という理由で] despising-is.
The frame [clause]+という理由で ('on the grounds that X / for the reason that X') is a productive way to attribute a stated reason to an action. ~という packages the clause as a label, 理由 ('reason') anchors it, and で marks it as the cause. ユーモアのセンス (literally 'humor's sense') is itself a calque from English 'sense of humor,' built from two English-origin loanwords (ユーモア, センス) — a small reminder of how English borrowings cluster in modern abstract vocabulary. The contracted 軽蔑してる (~てる for ~ている) gives a casual, spoken feel.