There is no question that good listening skills and a skill in demonstrating one's ability to listen are powerful tools in negotiation. Negotiation is an act of persuasion, and listening is at least as important as speaking.
To develop effective listening skills, one must practice, as one practices any skill. I would practice what I call mindful listening. This means emptying your mind of what you wish to say next and focusing on the person who is speaking. Most of us are very poor listeners because, while we hear what people are saying, we are usually too focused on what we are going to say in response, and this prevents us from taking in what we hear. Thinking about what you are going to say next is not really listening at all, is it? Look at the person who is talking to you, noting his or her face and body language, which is part of effective listening, too. As when you read, note key words, words that are emphasized or repeated. Repeat in your own mind the key words or concepts that stand out to you. This will reinforce your listening skills. If the context permits it, take some notes. For example, it is usually fine to take notes in a business meeting or a class. Taking notes helps you listen better, too, forcing you to focus on what is being said.
To demonstrate effective listening skills, there are many means. One is to "give back" to the speaker a brief summary of what he or she said. When we summarize what we have just heard and offer it back to the speaker, the speaker sees that he or she has truly been heard. This makes the speaker far more open to anything you have to say next. Feeling that we have been understood is something we all want. Look at the person speaking; meet his or her eyes. Do not behave restlessly and impatiently, tapping your fingers or swinging a leg. These are all cues that suggest you are not listening very well. Keep your body language "open," meaning not turning away from the speaker or folding your arms across your chest, which would show that you are not listening or, at the very least, are not receptive to what is being said. Nodding your head from time to time helps promote a sense of listening, too. Do not interrupt, except with brief encouraging words, the verbal equivalent of nodding your head, for example, "Okay" or "I understand." Most of us pause naturally when we come to the end of a thought, and you should allow the speaker to do so as well. There is a rhythm to this that you will feel when you practice.
Speaking skills are wonderful to have, but we are all called upon to be listeners, too. Be a good audience when others speak, first because it is simply good manners, second because you will learn something, and finally because listening is a persuasive act, in and of itself.