On the few occasions I've been through back in the days (before I knew game) where the girl did all the talking, it ended abruptly every time with her leaving me. I can understand that a guy who doesn't say much is perceived as boring, putting all the responsibility of the conversation on her. However, in my case the girl was literally babbling constantly without any pauses, so there was no room for me to say anything anyway, and I found the situation quite amusing.
The conclusion is that you need to take a more active part in the conversation. You don't have to lead it though if the girl takes that role willingly, since being a good listener is a good quality too. But you need to be involved, asking questions and relate to what she says and tell some stories of your own.