When you do something in a careful, trustworthy way, you do it responsibly. If you spend your money responsibly, you'll probably be able to save some of it.
Acting responsibly is important in life — it's what makes people trust you, and it means you usually do things only after thinking them over and choosing what's wisest or most sensible. You can't get a driver's license until you prove you can drive responsibly. And, before someone hires you to babysit, they'll want to know you act responsibly. The Latin root is respons-, "answerable" or "offered in return."