。
She finally managed to learn how to drive.
Literal
She [topic-は] somehow car [genitive-の] driving [subject-が] do-able [become-ようになる].
Two structural pieces work together. どうにか ('somehow, by some means, with effort') signals that the achievement came after struggle. The frame ~ようになる, attached to the potential form 運転ができる ('can drive'), expresses a change of state — 'come to be able to do X.' Together they paint the arc: it took effort, but she got there. The dictionary form 運転 plus が plus 出来る/できる is the canonical way to say 'be able to drive,' with が (not を) marking what one is capable of — a quirk of Japanese potential constructions.