Before Maariv, we say the pasuk "Vhu rachum yichaper avon..." (Psalms 78:38). Why?
Because sinners are lashed between minchah and maariv, and they say "Vehu Rachum" during the lashing (as per Makkot 22b), so the chazan also announces "Vehu Rachum" (Sefer HaManhig). Alternatively, because there is no korban tamid which maariv represents, but the tamid is supposed to atone, so we say "Vehu Rachum" instead (Pardes in the name of R' Eliezer HaGadol).
Sidur Rashi 425 - since they used to give Malkus to the sinners between Mincha and Maariv, therefore immediately after that we say Vhu Rachum to request Slicha & Mechila. It was said three times since there are 13 words in the Posuk. 3 x 13 = 39 the amount of Malkus.
Minhagei Yeshurin 31 - against the 3 angels (Malachei Chabala) that take care of Geheinom. Mashchis, Af, & Cheima. Vlo Yashchis is against Mashchis. Vhirba Lhoshiv Apo is against Af. Kol Chamoso is against Cheima.
In addition to the many good answers there is more discussion found here. I will quote one additional answer
They instituted the practice of reciting V’Hu Rachum before Tefilas Maariv. The reason to do is clear. In Midrash Ruth we learned that three entities rule in hell: Mashchis (destroyer), Af (anger) and Chaima (rage). They are responsible to judge the souls of the wicked. All others who serve there work under their control. Those agents of punishment create such a tumult that the sound can be heard in the heavens. Within the noise can be heard the voices of the wicked crying out: Oy Yoy and no one extends any pity towards them. These agents rule on each day of the week beginning at night and they judge those whose actions brought darkness and are forced in death to constantly think about what they did.
Therefore they instituted the practice of reciting the verse: V’Hu Rachum because beginning with the time of Tefilas Maariv the wicked are judged by the three agents, Mashchis, Af and Chaima. Notice that all three of the agents are mentioned in the verse: V’Hu Rachum. The words: V’Lo Yashchis represent Mashchis. Af is represented in the words: V’Hirbah Li’Hashiv Apo; Chaima, in the words: V’Lo Ya’Ir Chamaso. That is why we recite this verse on weeknights. Once Shabbos comes, the process of judging those in Hell pauses. The wicked in Hell are extended a rest. Once Kiddush Ha’Yom is recited on Shabbos the agents of punishment stop their work. That is why we do not recite the verse of V’Hu Rachum before Tefilas Maariv on Erev Shabbos. We do not want to stir the agents of punishment. It is prohibited to say the verse on Erev Shabbos out of respect for G-d who protects the wicked on Shabbos. We demonstrate that the process of judging the wicked has been suspended for the day and that the agents of punishment are prohibited from inflicting punishment on Shabbos
tl;dr the agents that administer hell beginning on week-nights have names that are found in that pasuk. We recite this pasuk to placate those agents of hell.
There is a mystical reason for this because that pasuk is associated with warding off malicious spiritual entities. Nighttime is more propitious for these forces. We do not want them to to intercede or cause damage to our prayer so we begin the evening prayer with this verse to keep those forces in abeyance. Source: Zohar II 135b.
We want to repel all the evil forces. There are represented in that Pasuk.
Source:HaRav Meir Eliyahu Shelit"a