The Mishnah Berurah writes, "One should be very diligent with Neilah because the goal of the Aseres Yemei Teshuvah is Yom Kippur, and the goal of Yom Kippur is Neilah. Everything depends on the close (הכל החיתום אחר הולך)."
Neilah is when Hashem signs the decrees for the coming year, and therefore, we should daven this tefillah with intense kavanah.
The Gemara (Yoma 86) says that chillul Hashem isn't atoned for until the day one dies. Nevertheless, the Meshech Chachmah explains that at Neilah one can receive atonement for chillul Hashem too.
The explanation is as follows:
The Yerushalmi explains that there is no forgiveness for chillul Hashem because no malach dares speak in his defense to say that the sin he committed isn't so severe. If he disgraced Hashem's name, chalilah, which malach will be so brazen to say that the sin isn't so severe? Therefore, when a chillul Hashem charge is brought to the heavenly court, all malachim remain silent, and none advocate on that person's behalf.
However, the malachim aren't part of the judgment process at Neilah. Hashem alone judges us, and when we plead before Him and beg forgiveness for the chillul Hashem we caused, He will forgive us.
The Ramchal writes that at Neilah, one can attain the level of Adam HaRishon before his sin. It is a time when we can become completely pure before Hashem. All sins can be removed. It is a time of immense compassion, and Hashem will forgive us.
G-t's Numen
The day after Yom Kippur is called G-t's numen, "Hashem's name." (נאמען ס'ט-ג). What is the reason for this title?
1) The Baal Shem Tov zy'a explains that for ten days, we've been saying הקדוש המלך, and now we go back to saying in shemonah esrei הקדוש ל"הא. Therefore, this day is called G-t's numen, "Hashem's name."
