They can do multiple things and it depends on the individual. The initial gains are usually from simply learning how to put more effort into throwing. My experience and what Bagonzi has related is that some pitchers pick up a couple mph after the first session with weighted balls and there's no way that is the result of increased arm strength. My experience is that after that initial increase progress is very slow as would be expected if building arm strength is what is helping increase velocity. Also, the studies tended to show that underload training alone was more effective than overload training alone. (I'm working from memory here and could be wrong.) That implies that learning to move the arm faster through underload training is more effective than increasing arm strength.
However, one has to remember that the studies were done with a group and each individual within the group had different results. That's probably why, as a group, a combination of underload and overload training worked the best. The people who benefitted most from overload training were getting that and the people who benefitted most from underload training were getting that, so the group as a whole improved more. It doesn't mean that a combination will help one individual more than underload alone or more than overload alone.
There's been speculation that if a player isn't throwing up to their potential due to a weak arm then overload training will help the most and if they've got the arm strength but just aren't able to move the arm fast enough then underload training will work best. I seem to remember that people had ideas for testing to see which was the case, but I don't remember what they were. It seems like the test would be making throws with a weighted ball and if your velocity drops significantly then it is probably due to lack of arm strength. If it doesn't then you've got the arm strength but need to learn how to move the arm faster and underload training is indicated.
My recommendation is that if one is willing to take the risks that there may be with weighted balls to go ahead and do a combination of overload and underload training because you aren't going to know ahead of time which will work best for you.
Back in the distant past we intended to have my son do a weighted ball program going into his junior season to try to add that 2 or 3 extra mph. However, with all the injuries he's had our goal is more to get him healthy than to worry that much about increased velocity so we're simply going with as safe of a long toss program as we can. Personally, I think that long toss is a good way to increase velocity, but there isn't hard proof of that, so I'm just going with what I think will work. Perhaps if he stays healthy all season this year we'll give the weighted balls a shot going into his senior season.