The proverb is 'a rolling stone gathers NO moss'. It's not glaringly obvious what it means: I've always taken it to mean that someone who's always on the move, changing jobs, changing houses etc will never be materially successful.
Personally, I think the 'gathers no moss' element means that the wanderer avoids responsibilities and commitments...ie is always free and unhindered, with nothing to tie him down, as it were. Remember the words in the Lee Marvin song in 'Paint Your Wagon'..."I never saw a place that didn't look better lookin' back"...or some such phrase.
I always thought it meant a person who remains idle can become...stale (for lack of a better word). Activity keeps you...fresh (for lack of a better word). I wish I could think of better words, but get the idea?