I think by explaining why you are lying. You do not need to explain too much detail if it is more painful, but you have to be honest about why you are lying. Maybe you want to protect him, maybe you know that the truth will hurt him, or maybe you are ashamed of telling the truth and want to cover it up. Whatever the reason, let him know why you did it. This will show that you can grow and try to do self-introspection, and you are not impressed just to apologize without getting a valuable lesson.