The custom is to begin Selichos on a Motzei Shabbos. Leket Yosher (Orach Chaim vol.1 p.118) explains that people are happy on Motzei Shabbos due to the oneg Shabbos and the Torah they learned. A happy mindset is the correct mood for saying Selichos.
We say in the Selichos, שיחים נשפוך פנים באלה. The literal translation is, "With what face can we pour out our words?" This expresses our shame to speak before Hashem due to our many aveiros. However, we can translate it as a statement: פנים באלה, with the face we received on Shabbos, שיחים נשפוך, we can pray before You. This is based on the Midrash (Bereishis Rabba 11:2) ויקדש השביעי יום את אלוקים ויברך אותו, "'Hashem blessed the seventh day...' This means that Hashem blessed them with the shine of His face, and Hashem sanctified them with the countenance of His face.' On Shabbos, our faces shine with joy; with this face, we can approach Hashem and beg for forgiveness.
Rebbe Yissachar Dov of Belz zt'l gave another reason why we begin saying Selichos specifically on Motzei Shabbos. It is based on the Chazal (Shabbos 119:) that says, "Whoever says ויכלו [at kiddush], becomes a partner with HaKadosh Baruch Hu in creation." A partner has permission to express his opinion. For example, a business partner can say how he wants the business to be run. So too, a partner in creation has a right to give his input on how he wants the world to be run. After Shabbos, after we became a partner with Hashem in creation, we have a right to express how we want Hashem to lead the world, and as a partner, Hashem will take our requests into serious consideration. We are partners, and our opinions matter.
We can also explain that we begin saying Selichos after Shabbos because in the merit of Shabbos, our aveiros are forgiven. The Siduro Shel Shabbos says that ת"שב is roshei teivos for שובת וב בתש – on Shabbos you shall repent. The Chozeh of Lublin zt'l (Divrei Emes, Matos) writes that he heard from Rebbe Elimelech of Lizhensk zt'l that Shabbos is the shoresh (origin) of teshuvah. Indeed, the translation of שבת is to return to Hashem.
It states (Yeshayah 42:18), והעורים שמעו החרשים לראות הביטו, "Deaf ones, listen! The blind, see!" The Chidushei HaRim zt’l asks, how can the deaf hear? How can the blind see? He answers, "There are moments so lofty that even the deaf hear and the blind see." Shabbos Selichos is such a time, when everyone is inspired with a holy urge to improve their ways.
10. When Rebbe Yissachar Dov was old and unwell, his gaba'im tried to convince him not to go to beis medrash for Tehillim on the first day of selichos. But he came to the beis medresh with mesirus nefesh. He said, "How can you prevent me from the great hisorerus and hislahavus of saying the רצון יהי after Tehillim together with the congregation?"
