it depends on how far you've gone in your relationship
I guess some people just don't want to talk a lot when they are not familiar with others. I mean if you don't want to talk to the girl when you think you should have talked, you actually don't know her very well or you're not sure about how much to reveal. For example, you're not sure how far you've gone in your relationship with her, or you're not sure what her reaction would be if you say this or that. I think everybody has a lot of story to tell, just that they don't want to tell or don't think these stories worth telling, that's how people get quiet.
Not every girl like talkative guys. I know a lot of guys are really talkative, but I think interaction is more important. Let the girl to know you really cares about her feelings and reactions. You don't want to talk through your date with her saying nothing, right?