The patient was rushed to the hospital by ambulance.

Literal

Sick-person [topic-は] ambulance [by-で] hospital [to-へ] urgently was-carried.

救急車で uses instrumental で ('by ambulance'). 急いで運ばれた combines the te-form adverbial 急いで ('hurriedly') with the passive 運ばれた ('was carried'). The passive makes the patient the grammatical focus, natural for describing someone being transported.