。
He's not really the type to enjoy jazz.
Literal
He [topic-は] jazz [object-を] enjoy-cultivate [like-ような] type.
たしなむ is a somewhat refined verb meaning 'to enjoy (a cultured pursuit) as a hobby, to be practiced in' — often paired with refined activities like tea ceremony, jazz, poetry, moderate drinking. ~ような is 'the sort of ~, the kind that ~,' used to characterize a type. タイプではない = 'not that type' with polite/neutral formality.