The etymology of care starts with Old English, caru, cearu "sorrow, anxiety, grief," also "serious mental attention." Carefree would mean a state of being without sorrow, or serious mental attention.
Yesterday I cycled into work, and on the way home, as the rain began to fall, I had a feeling that I don't think I had had for a very long time, which I can only describe as carefree. It is a state that I would like to foster as an constant way of being.
Carefree is not a state of laziness or irresponsibility. I have a family to provide for, students to teach, and a farm to attend to. I responsibly fulfill my roles every day.
Carefree is a state of mental freedom. It is a condition of fearlessness. Carefree is a choice one with a free mind can make.