When people are described as "batting their eyelashes," they are fluttering their eyes open and closed rather quickly in order to attract attention. This is supposed to be a seductive, alluring or flirtatious gesture, though it can also be the subject of humor or mockery as well.
The verb "bat," when applied to eyelashes, refers to the fluttering motion created by long eyelashes connected to eyelids that are rapidly opening and closing. The gesture is somewhat subtle and doesn't require people to open their eyes very wide. Intended as a flirtatious gesture, the sincere batting of the eyelashes is not a common occurrence in modern society, though it is referenced in stories.