When you hit a baseball with a bat, how is momentum transferred?

When the bat hits the ball, it exerts some force on the ball. Just think about a home run hitter hitting a stationary ball. How far do you think it will go? Will it go more than 400 ft.? Probably not. While the kinetic energy transferred from the bat to the ball accounts for some energy of the ball, it does not account for all. Where is the mysterious energy coming from?
The answer is conservation of momentum.
I just said momentum is conserved but how do I know that? I know that because of Newton's 2nd law: F=ma (Force equals mass times acceleration)
Conservation of momentum means that the harder you throw you, the harder the ball will bounce back at you. That is the reason it is easier to hit a home run on a fast ball than a curveball.
Conservation of momentum also means that the bat can transfer some of its momentum to the ball. This is why it is better to use a heavier bat if you swing just as fast. The momentum is the product of the mass and velocity, so to make it easier to understand; a heavier bat swung at the same speed as a lighter bat will have more momentum.