The expression comes from the saying: "A rolling stone gathers no moss." This means that someone who frequently moves from one thing to another does not put down roots, develop life experience, and does not have a serious attitude to life. So a "rolling stone" usually has these negative connotations. A fairly close synonym is "drifter".
Perhaps there are also situations where it has a more positive connotation of freedom - I can go wherever I want, whenever I want. Context is key but probably the meaning in your lyric is the negative one.