。
She quickly slipped her arm through his.
Literal
She [topic-は] her own arm [object-を] swiftly his arm [into-に] caused-to-slip-in.
すべりこむ (滑り込む = 滑る 'slide' + 込む 'enter into') is a compound verb meaning 'to slip into, to slide into.' Its causative すべりこませる ('cause to slip into') reframes the action as a deliberate, deft motion — she didn't passively let her arm slide; she threaded it into his. The image is of slipping one's arm into the crook of someone's elbow, a small physical gesture of intimacy or affection. すばやく ('quickly, deftly') tunes the manner of the action.