they want it. There isn't really a logical one to make like "pitchers can't hit" or anything like that it is really just a matter of they prefer it and that is fine. I have found most guys that love the DH are AL fans and most guys that don't are NL fans. I am a Dodgers fan so I fit the stereotype, my brother and one of my buddies are Red Sox fans and they love the DH so they also fit the stereotype.
One of the things I really like about pitchers hitting is that when a pitcher gets pulled it generally either enables or forces a manager to use, and of course burn" a pinch hitter. There is more strategy in the move than what you see in the AL where it doesn't affect the lineup or the bench at all other than the bullpen. In the NL it affects the bullpen and the bench and is just another chess move that makes things even tighter as the game goes on. You seldom ever see the double switch in the AL but you see it almost every game in the NL. Just more things for the manager to actually manage, I like that.
But like I said I don't hate the DH, it's fine having it in the AL but I prefer to not have it in the NL. Interestingly it is the guys that love the DH that make more of an issue about the NL not having it than guys that don't want the DH making an issue with the AL having it. I think it is because the AL/DH guys feel like they are at a disadvantage playing in NL parks in the WS because their pitchers who haven't had to hit all year, and maybe ever, have to take their at bats while the NL pitchers have been hitting all season and are better at it and more likely to contribute with their bats than the AL pitchers.