I use to talk too much about myself. My wife would nudge me or give me a "shut up, Dave" look. Sometimes I still find myself doing this, usually when she's not around. It reminds me of the classic RUN DMC song. What I have learned is that asking open-ended questions is always the way to go when meeting new people. Avoid Yes or No questions to allow the conversation to blossom.
Listen. I can't stress this enough. Listen, and perhaps you will learn how you can help them with a problem or task. More on that here.
If you can help then and there, great! If not, make a note on the person's business card. Be sure to follow up with the assistance they are seeking.
I get asked for Linkedin introductions all of the time. This is a great way to help connect your network.
Always look for ways to help people. Can I help you?
Photo from Flickr by: library_mistress