What is the importance of being responsible?

Responsible behaviour, primarily, protects you and the people and things you care about, from harm or other negative change. Teaching your kids to always wear a seatbelt, might one day safe their lives; wearing a condom if you're sleeping around can safe you from getting HIV or other nasty STD's; and regularly maintaining appliances you own will prevent their premature breakdown.

A secondary plus side to being responsible, is that you get to be responsible over things, which - if you are responsible enough - will make people believe you are a responsible person (which you might very well be) and they will trust you, perhaps with even more responsibility. You can see how this works well within companies, where people who show managerial capacity (mainly being responsible and relatively good with people) rise on the company ladder, being given more and more people and processes to oversee; and more and more responsibility.

So people will trust you.