Fenway is one of the relatively few major league ball parks that allows us to bring in big lenses. They take a very reasonable approach: don't bother the other fans and we won't bother you. I wouldn't try to bring in a monopod, but a 100-400L at full extension with lens shade: no problem.
As to getting those action shots of the ball on the bat, it's largely a matter of luck. During the game, each time a Red Sox batter was at the plate, I zoomed in on him, framing him in the middle and putting the umpire and catcher way to the left of the frame. Just as the pitcher winds up, the batter always tenses up and brings his bat back, getting ready to swing. At that instant I press and hold the shutter release. I shoot at high-speed continuous, taking only high resolution raw images. I also set the autofocus/drive menu setting on "2: Release/Drive speed priority" to maximize my shutter rate. By pre-focusing on the batter (using only back button focusing), the camera doesn't try to change the focus as the shutter clicks.
Even with all these things, it's still a matter of luck. The 7D can shoot at a maximum rate of 8 frames per second. In 1/8 second, the ball can enter the image frame, get hit, and leave the image between shots and never show in a photo. My solution: take photos during just about every pitch. Out of about 800 images taken during a game, I might get 3 or 4 with the ball on the bat. The day I took the shots above I was lucky, getting about 5 or 6 good shots.
The second image (duplicated in the fourth) shows Travis Shaw getting his first ever major league home run. I really should contact him and offer him a copy.