Here's a good MLB Diamond Demo with former Royal's All-Star Mike Sweeney talking about vision in the batter's box. First, he discusses a fairly common sense mechanical aspect of keeping the head down on the ball: During the course of the swing, the batter's chin should go from the front shoulder to the back shoulder. If both eyes are seeing the ball as it is pitched, and the hands load properly before pulling forward, the chin will touch the front shoulder as the ball is approaching. If the eyes track the ball in to contact, and the hands finish high enough in the follow through, the chin will then be touching the back shoulder.
If the chin doesn't go to the back shoulder, the batter is not only not seeing the ball as well as he should. But he is also likely pulling his bat away from the ball, instead of to it and through it. The head is obviously connected to the neck, then the shoulders, then the arms, hands and bat. I like to think of the bat as an extension of the head. If the head is down and in at contact, there is a better chance of meeting the ball with the barrel of the bat. If the head flies out, there is more of a chance of weak contact at the end of the bat - if there is contact at all.
Sweeney also discusses the importance of the batter focusing on the ball as it is being pitched. As they say - "see the ball, hit the ball." To me, this means the batter needs to clear his head of any thoughts apart from the baseball itself when he is preparing to swing. His analysis of the previous swing, his necessary adjustments, his emotions, his planning - that all needs to happen before he digs in. Once he sets up in the box, the batter can rest assured that he is prepared. He can take a deep breath, relax and focus on the ball. With this simple, relaxed focus, there is a better chance of a timely, mechanically sound swing - and therefore a better chance of a good hit.