。、、、。
She has many hobbies — cooking, knitting, gardening, stamp collecting, and so on.
Literal
She [topic-は] many [genitive-の] hobbies [object-を] is-having. Cooking, knitting, gardening, stamp-collecting [etc.-など].
持っている here is the resultative ~ている of 持つ ('hold / have / possess'); it covers 'possesses, has' for ongoing ownership. The second 'sentence' is an unframed list — a stylistic device where examples follow the main statement as a fragment, closed off with など ('and so on, etc.'). The exhaustive listing of hobbies relies on commas (touten 、) without conjunctions; Japanese lists work this way both in spoken and written text.