She grows flowers like tulips, pansies, and daisies.

Literal

She [topic-は] tulips, pansies [and others-や] daisies [like-のような] flowers [object-を] [is-growing-育てている].

や is the non-exhaustive listing particle: 'X, Y, and others (of similar kind)' — distinct from と, which lists exhaustively. The frame [examples] + や + [last item] + のような + [head noun] is a stock way to list typical members of a category before naming the category itself: 'tulips, pansies, daisies, and flowers like those.' Note the comma after チューリップ — Japanese also uses commas in lists, alongside particles, when mixing structured listing with prose. 育てる ('raise, grow, cultivate') covers raising children, plants, animals, even talents.