I can suggest three things three things Pouncer. The first has two parts. First, examine your intentions. Take 15 minutes alone and think about why you tend to interrupt people as they speak. Really nail it down. Is it in fact because you are "afraid of people opening up" to you? If so, what are you afraid of? There's allot to consider here so give yourself enough time to understand this.
The second part of examining your intentions is to focus on your motivation is for any specific conversation. Do you want to get to know someone better? Look good? Impress someone you want something from, such as a job, sex, a business deal. Do you perhaps want to help them? When you examine (and clearly understand) your intentions you will come to understand yourself a bit better and become a little more focused.
Second use reflective listening. People like confirmation that they are being understood. You can say things like "Let me make sure I've got this right, you're saying XYZ. Correct?" It just shows you are listening and understand what is being said.
Third ask questions about the person you are speaking with. Asking questions shows you are concerned about that person. It also shows you don't need to be the center of attention. Some of the people that I care most about routinely ask me the most pointed questions. The care they have shown by doing so is a fundamental part of an ongoing good relationship.