What does "vulnerable" mean in this context?

Exactly what the dictionary says: open to attack or injury of a non-physical nature. It doesn't sound very good when put in those terms, granted, but the main idea is that showing vulnerability is a way of signalling trust. You give someone the power to harm you, but you trust them not to abuse it. One form is sharing secrets or personal details (if you read HPMOR, this point comes up).