I have come to know that touching the private parts invalidates the Wudu. Is it direct skin-to-skin contact, or even if the private parts are touched over the cloth that invalidates the Wudu? Sometimes it happens while getting up in Salat, the testis gets stuck between the legs and one touches them to get them out; does this invalidate the Wudu?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Touching the sexual organ (penis-vagina) with the palm of the hand and without any barrier (clothes and so on) invalidates the Wudu.
But, if there is some barrier between the organ and hand, this does not invalidate the Wudu.
As for touching the testicles, this does not break the Wudu, whether one touches them with/out a barrier. Touching the anus follows the same ruling although al-Shafi'ah and al-Malikiyah are not of the same opinion.
Allah knows best.