The first time —and the last— I tried to play baseball I injured my shoulder trying to throw the baseball from the right field with all my strengths and my hips trying to hit.
It looks easy but it is not at all. You need a lot of practice and perform properly with the required and adequate technic —even when you throw.
Said that, MLB players are baseball beasts. They've been playing the game since forever. Even if they are well trained injures might come. Resting people won't avoid latent injuries. This game does not demand exhaustive physical performance all the time like other sports.