I work with student leaders every year who are preparing for competitive events where they must answer questions from judges - pageant contestants, competitive speakers, etc. The following strategies are the bedrock of everything we work on together to help them shine answering questions.
1. Talk in a conversational tone. The Q&A portion is essentially a formal conversation. Without being too casual (using slang, filler words, etc.), be conversational with your answers. Change the pace of your voice. Look the judges directly in the eye. Don't worry about saying the exact right thing. If you say something that needs adjusted, do so. Back up and say it again the right way. If you need to, talk about the process. IE - "that is a great question", "I have studied for many hours, but the answer to that question is escaping me at the moment."
2. Answer in list form. This strategy is especially important if the judges listen to multiple people in a row. IE - they are in information overload mode. This is also a valuable tactic if you tend to ramble during your answers, which can happen to even a seasoned pro. List form is exactly what the name implies. As the judges are asking their question, you quickly think of two or three answers. When you begin your answer, you say something similar to, "There are three great ways to answer your question." And then you take them through the three answers. This is effective because it gives your answer structure (to keep you from rambling) and it helps the judges have a clear and concise method for following your answer.
3. Use an anchor word. If the answer doesn't demand length, try the anchor word strategy. This is where you begin with one word that most accurately sums up your answer. Then you give one or two support points about why you chose that word. Most times simple and brief will be scored better than detailed and extravagant.
4. Tell stories. I have served as a judge at many competitive events and the most impressive tool I have seen students employ is telling stories in answers. This is impressive because it demonstrates a depth of knowledge, an attention to detail and it gives your answers faces and places. Remember to give your story meaning by briefly connecting the story to their question.