Torah Temimah (citing the Rambam in Peirush hs'Mishnatos): Hashem gave them jurisdiction over the matter, because they witnessed the act first-hand, whereas everybody else only heard it from them.

Oznayim la'Torah: Because having heard the sinner specifically state that he will ignore their warning and contravene the Torah's instructions, and seen him putting his words into practice, that inevitably creates the impression that either the sin is insurmountable or it is not so terrible to transgress it, an impression that needs to be eradicated - and that can only be done by punishing the sinner. ¹

Sanhedrin, 45b: The Pasuk implies that they must put the sinner to death with the same hand that they had when they testified. Consequently, they become Pasul and the sinner is Patur from Misah in the event that their hands were severed after they testified, ¹ but not if they had no hands already before they testified.

Sanhedrin, 45a: One of the witnesses pushes the sinner down - from a height that ie equivalent to two men; in the event that he is still alive, the second witness (with the help of the first one - Sanhedrin Ibid.) drops a stone on his heart. ¹ In the event that he has still not died, the people stone him to death.

Rashi (in Chulin, 139a): It implies that it is a Mitzvah for anyone who comes across someone who is Chayav Misah, to take him to Beis-Beis in order to destroy the Resha'im from Yisrael.

Sifri: It teaches us that Beis-Din are obligated to destroy evil from Yisrael. ¹

Refer to 19:19:6:2.