What did it mean, when she stared at me?

To start with you need to make yourself more appealing to this girl. She has a lot of friends and you have very few. That is not going to get her to go out with you. I know you are shy but you must find a way to get closer to her. I would suggest trying to make friends with one of her friends and try to get invited places she will be. I am not sure what to say about the stare. What you are describing sounds more like a daze. Sometimes at meetings I saty at someone or something without realizing it becaus ei am bored.