I believe CoachPaul is referring to getting hit by a pitch in your throwing arm/hand when hitting opposite. When you hit same-handed, your throwing arm is your back arm when you’re in your batting stance. But when you hit opposite-handed, your throwing arm becomes your front arm so it is more exposed.
But there is another way a pitcher’s throwing arm can get damaged when hitting opposite. Hitting is a joint loosening activity for the front shoulder. After the bat passes through the contact zone, it wraps around behind you during the follow-through which opens up the front side of the front shoulder and, over time, can loosen the shoulder joint. This may not be a problem for pitchers who bat same-handed because it is their glove side shoulder that is getting loosened. But, for pitchers who bat opposite-handed, this can be problematic because, obviously, it is their throwing shoulder that is getting loosened - especially for those who release the top hand from the bat during follow-through. Unfortunately, the problem might not show up until years down the road.