Why does this girl keep staring at me and smiling?
She could like you! Next time you catch her staring at you, smile back. If she looks down, and looks away, that means she's shy around you, and that you've realized her staring at you. Most likely she's a little to shy to talk to you, so why not go up and chat with her. It doesn't have to be something she likes, if she likes you, then she'll be interested in what ever you've got to say.
Probably because she likes you and wants you to start talking to her
Maybe: She likes you. She is amazed by you. She is interested by you she is amused by you she really likes you she has a crush on you etc.
Sounds like you are the target of a inside joke. If she keeps staring and smiling, something is up. If she was bold, she'd say something. If she was shy, she'd try not to be caught staring at you.