The importance of being responsible is that it shows others that you are a trustworthy person. People might give you more responsibilities if they see that you protect and care about the people and things around you.
Being responsible is important because it means you are accountable for your actions and you are trustworthy and people can depend on you.
Responsibility is important because some things have to be done whether or not you want to do them. If you are responsible, you understand your obligation to take care of things.
It is important to be responsible because by choosing to be responsible you improve your life. If you act responsibly then people will learn to trust that you will do what you say