To emphasize is to make something important, or stress it, like when you were little and your parents would always emphasize the importance of looking both ways before crossing the street. They told you again and again and again. And again.
The verb emphasize works when you really want to stress a point or give extra weight to an issue. Emphasize comes from words that mean “significance” and “to present or show.” A player who finally makes the varsity team might emphasize this achievement by always wearing his practice jersey — even off the field, or you might emphasize the importance of being on time by locking the door on latecomers.