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Make sure that no one is angry or emotional beforehand and that you have plenty of time to talk about any issues. Don't bring this up in front of younger siblings. You might make notes for yourself, so you don't forget to touch on every point. Explain to them exactly why you think you can handle this particular freedom, and don't raise your voice. Becoming angry or upset will only prove to them that you are not yet mature enough to handle additional freedom. Identify which specific freedoms you want. This might be a later curfew, more computer time, or permission to go to a concert. Be prepared that this might be an ongoing conversation. They may need to think about it before making a decision. Talk through any concerns that they might have. For instance, if you're asking to go out with a friend, see if you can get them in contact with that friend's parents. Remember to remain calm, even if they say things you disagree with. Think about any possible risks or consequences, and be ready to answer any questions your parents might have. Perhaps you get decent grades consistently, do your chores without complaints, and interact well with your siblings. Remind them of how responsible and hardworking you've been acting recently. If you've done anything particularly special, such as cooking dinner for your parents or taking on more chores, be sure to include that as well.
Choose an appropriate time to discuss this topic. Ask them in a calm and mature manner. Consider their point of view. Give recent examples that prove that you are dependable.