He won three super bowls (92,93,95), plus a college football championship (while injured), in 1985.
He wasn't a member of any coaching staffs who won championships, as far as I can tell - he's been a commentator since his retirement, for Fox.
From looking at the picture, there are three superbowl rings, which would be his pinky finger, middle finger, and thumb. There is one Pro Football Hall of Fame ring, which is the lower ring on his ring finger (the oval-shaped purple ring). One would be his College Football Hall of Fame ring, which is likely the other ring finger ring. The sixth could be one of a few things; it could be his college football championship ring, although that should be red. It could be a ring for an MVP or some other honor, or possibly for the Dallas Cowboys Ring of Honor, though I didn't see any evidence of that existing.