Why is moisture a chatzizah for netilas yadayim? I guess you can say that the hands must be completely clean, but then why don't you have to wipe your hands in middle? For example, one who washes his hands twice right and twice left, he should wipe his hands in between the first right and the second right.
It's not a matter of a chatzitzah - anything liquid wouldn't be a chatzitzah, because the water will wash it away (Shulchan Aruch, Orach Chaim 161:2).
It's rather that the water from the first washing becomes tamei by touching your hands (which themselves are tamei - that's why you have to wash them), and the second washing then renders the hands, and the remaining water from the first washing, tahor (ibid. 160:2,4). So there's no reason to dry your hands between washings.
(If your hands weren't clean before you started washing, then you need three washings: one to remove the dirt or whatever, and then a second and a third following the same principle as above. There are halachic opinions that three washings are always needed; this is the Chabad custom.)
It is actually a machlokes.
Rabbi Kaganoff discusses the subject in This Is the Way We Wash Our Hands and during that posting gives several reasons that would lead one to use a cloth when holding the cup.
Rabbi Kaganoff explains that there is a machlokes as to whether the hands must be completely dry at the beginning of the washing and if the cup itself must be dry as well. That is why some people will hold the cup with a towel or carefully dry the hand that is about to be washed. Perhaps the chassidim that you speak of follow the psak of the Chazon Ish that Rabbi Kaganoff cites.
Washing wet hands
Must one’s hands be completely dry before you begin washing netilas yadayim? The authorities dispute what the halachah is in this case.
As we learned above, someone who, when pouring water for the first time, rinsed only part of his hand, must dry his hand thoroughly and begin the procedure over. The authorities dispute whether one must always have dry hands when beginning netilas yadayim or whether one may perform netilas yadayim even though his hands are wet or the handle of the cup is wet. According to the Magen Avraham (162:10) and the Mishnah Berurah (162:27), one may begin washing netilas yadayim, even though one’s hands are wet. The Chazon Ish (Orach Chayim 24:20) disagrees, contending that one’s hands must be dry when one begins washing netilas yadayim. Therefore, the handle of the cup must also be dry or, alternatively, one may grip the handle of the cup with a towel or some other item that keeps his hands dry until he washes netilas yadayim.
Another possibility is that the water on the cup may be considered used water, which can cause problems. Note that if the first hand is held properly, the water from the first hand being washed will not flow onto the cup so there would be no problem of tum'ah.