。
She walked clinging to his arm.
Literal
She [topic-は] his [genitive-の] arm [to-に] cling-while walked.
~ながら ('while') attaches to a verb's masu-stem to mark a simultaneous action. The main verb (歩いた, 'walked') gives the primary action; the ~ながら verb (しがみつき, 'clinging') gives the accompanying action. しがみつく ('cling, hold tight') takes its target with に. The image is intimate — physical proximity expressed through the held arm, the simultaneous walking — common in romantic or anxiety-driven scenes.