If you live out a dream or idea, you do the things that you have thought about. He began living out his rock `n' roll fantasy during his last year in law school I suppose some people create an idea of who they want to be, and then they live it out
If you live out your life in a particular place or in particular circumstances, you stay in that place or in those circumstances until the end of your life or until the end of a particular period of your life. Gein did not stand trial but lived out his days in a mental asylum I couldn't live my life out on tour like he does