It is customary to begin the Arbit prayer service with the recitation of the verses, "Ve’hu Rahum Yechaper Avon Ve’lo Yash’hit Ve’hirba Le’hashib Apo Ve’lo Ya’ir Kol Hamato; Hashem Hoshi’a Ha’melech Ya’anenu Be’yom Kor’enu." The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Pekudeh (Shana Rishona, 1; listen to audio recording for precise citation), explains that these verses are recited in order to counter the forces of harsh judgment which surface during the nighttime hours. As the Arizal (Rav Yishak Luria of Safed, 1534-1572) noted, the verse of "Ve’hu Rahum" has thirteen words, which correspond to the thirteen divine attributes of mercy. We thus recite this Pasuk to arouse G-d’s compassion and thereby negate the effects of the forces of judgment. Moreover, the Arizal taught that some of these forces are called "Mash’hit," "Avon," "Af" and "Hema," and in this Pasuk we ask Hashem to oppose these forces so they do not harm us. We then recite the verse, "Hashem Hoshi’a," asking that G-d should save us and protect us from harm.
For this reason, the Ben Ish Hai comments, we recite "Ve’hu Rahum" before Arbit only on weeknights, and not on Shabbat. On Shabbat we are granted protection from harmful spiritual forces, and thus the Zohar, as the Ben Ish Hai cites, writes that it is forbidden to begin our prayers with verses relating to the topic of harsh judgments. Although we recite "Ve’hu Rahum" in our Zemirot on Shabbat, this is because we are reading verses of Tehillim. But it would be inappropriate to introduce a prayer service on Shabbat with this verse, given the special quality of Shabbat which protects us from harsh judgments. Therefore, we recite "Ve’hu Rahum" at the beginning of Arbit only on weeknights, and not on Shabbat.
Summary: The verse of "Ve’hu Rahum" is recited at the beginning of the weekday Arbit prayer service because it has the power to oppose harmful spiritual forces which surface during the nighttime hours. On Shabbat, however, when these forces do not surface, we do not introduce Arbit with the recitation of this Pasuk.