Winnie wanted to run away because she was bored.
Winnie didn't exactly run away; she was kidnapped. Winnie feels smothered at home to the point that she “lost her patience at last and decided to think about running away” (Prologue). She lives with her mother and grandmother, who like everything neat and tidy. Winnie feels all their focus is on her because she has no siblings. Things at home are too stifling, prompting her to say,
It'd be better if I could be like you, out in the open and making up my own mind. Do you know they've hardly ever let me out of this yard all by myself? I'll never be able to do anything important if I stay in here like this. I expect I'd better run away (Chapter 3).
Winnie isn’t really sure she should run away, as she doesn’t have anywhere to go. She may not have had any adventures or done anything exciting, but that is about to change. One day, Winnie is in the woods and she sees a boy drinking from a spring. When she tries to drink from the spring too, the boy does not know what to do. He can’t let her, because she will become immortal if she drinks from that spring. To explain everything to Winnie, he and his mother decide to take her home with them.
Afterward, when she thought about it, it seemed to Winnie that the next few minutes were only a blur. First she was kneeling on the ground, insisting on a drink from the spring, and the next thing she knew, she was seized and swung through the air, open-mouthed, and found herself straddling the bouncing back of the fat old horse (Chapter 6).
Winnie had an adventure after all—maybe more than she bargained for. She knew her family would be worried and she was homesick, since she had never been away from home. Winnie really did like the Tucks, though. They seemed like very nice people and not at all like kidnappers. They just wanted to make sure she fully understood her situation.