In the story, Arnold threatens Connie into going away with him despite her intense resistance. Early in the story, he tells her, "Gonna get you, baby." Initially, Connie thinks nothing of it. She assumes Arnold is another harmless admirer. However, Arnold turns out to be a major danger, one that might potentially end her life.
Arnold is privy to information he should not be, such as where Connie's family is the day she's home alone and even what her older sister is wearing at the picnic. He demands Connie get into his golden car. When Connie threatens to call the police on him, Arnold claims he will stop her physically if she even tries and that he will hurt her family if she continues to resist his demands. Ultimately, Arnold is able to coax Connie to him with his threats, using her love for her family as a way of forcing her hand.
As to what Arnold does to Connie after he lures her to his car, the story is not explicit. Rape and murder seem inevitable from Arnold's sexually threatening comments and Connie's sudden premonition that she will never return home or see her family ever again. However, Oates leaves Connie's ultimate fate uncertain.