Taking responsibility means that you are in control of the actions that happen in your life, that you are in control of your own sadness or happiness, your success or failure. If you cannot take accountability for your own actions, then it will be hard for you to grow as a person.
Being responsible and taking responsibility are two related but different concepts. Just because you are a responsible person in the actions that you take does not necessarily mean that you take responsibility for the actions you do wrong. Owning up to your mistakes is an important milestone in growing up. Everyone makes mistakes; it is unavoidable. However, it is how you handle your mistakes and learn from them that will help you grow as a person and help to build your character.