The kaparah of Yom Kippur extends for another four days. This is one of the reasons we don't say tachanun during these intermediate days.
Magen Avraham (624:7) states that we don't say tachanun because Shlomo HaMelech celebrated the Chanukas Hamizbeach for the first Beis HaMikdash on these days (see Malachim 1, 8:65). But it is also because we celebrate the kaparah that extends from Yom Kippur into these days.
The Shlah HaKodesh (Succah, Torah Or, 2) explains that on these four days, people are occupied with mitzvos, to buy a lulav and esrog and to build a succah, and their preparations for the mitzvos, performed with a holy desire to serve Hashem, cleanse them from all aveiros.
We will bring some sources for our atonement on these four days. Prominent among the sources is the pasuk (Vayikra 23:40) הראשון ביום לכם ולקחתם, "Take [lulav] on the first day." Why is it called the first day? Succos is on the 15th day of the month! The Midrash (Tanchuma Emor 22) explains that the 15th of Tishrei is called הראשון יום because "It's the first day for reckoning aveiros..." Until Succos, we still have the kaparah of Yom Kippur.
Another source is from the Gemara (Yoma 20.), which says that השטן is gematria 364, one number less than the days of a solar year. השטן, which is gematria 364 implies that the Satan can prosecute on 364 days of a solar year, but on one day of the year, he must remain silent. He can't prosecute on Yom Kippur.
But notice that the Gemara discusses השטן (which is gematria 364), but שטן, without the heh as its prefix, is gematria 359. Reb Yonason Eibshitz zt'l (Yaaras Dvash 2:10) writes that this indicates that there are six days a year that the Satan can't speak against us. These are Yom Kippur, the four days between Yom Kippur and Succos, and the first day of Succos. This hints at the purity from aveiros we merit during these days.
The Gemara (Shabbos 119) says, "The world exists in the merit of the Torah study of cheder children. Rav Pappa asked Abaya, 'And what about my Torah and yours? Isn't our Torah sufficient to give life to the world?' Abaya replied, 'You can't compare the voice [of a child] who never tasted sin to the voice of someone who tasted sin.'" Young children have never experienced the taste of sin; therefore, their Torah is so precious to Hashem, and their Torah study sustains the world.
The Kozeglover Rav zt'l (Eretz Zvi) writes that during the four days between Yom Kippur and Succos, even an adult’s Torah study sustains the world because on these days, he is clean from sin. It is even a higher Torah than a child’s Torah because an adult is a ועושה מצווה, obligated to study Torah, while a child isn't.
The Chasam Sofer zt'l was in an extremely happy mood between Yom Kippur and Succos because of the atonement, to the extent that he couldn't focus on studying Torah. So, on these days, he composed songs and poems to express his yearning and gratitude to Hashem (collected in Sefer Shiras Moshe). We aren't on that level; we can study Torah after Yom Kippur, and we should! And our studies at this time sustain the world.
It states (Yeshayah 55) בהמצאו 'ה דרשו, "Search for Hashem when He is near." The Gemara (Rosh Hashanah 18) asks, "When is Hashem near? הכיפורים ליום השנה ראש שבין ימים עשרה אלו, 'These are the ten days between Rosh Hashanah and Yom Kippur.'" The Shlah HaKadosh (Succah, Torah Or 70) asks that there are only seven days between Rosh Hashanah and Yom Kippur. Why does the Gemara say there are ten days?
The Shlah answers that the Gemara hints at the ten days between Yom Kippur and Hashanah Rabba. These are the days that Hashem is nearby. Yom Kippur is called Rosh Hashanah, and Hashanah Rabba is called Yom Kippur, a day of atonement. Between them, there are ten days – days to utilize to "Search for Hashem when He is near."
