The kaparah of Yom Kippur extends for another four days, which is one of the reasons we don't say tachanun on these intermediate days.
Magen Avraham (624:7) states that we don't say tachanun because Shlomo HaMelech celebrated the Chanukas HaMizbeiach for the first Beis HaMikdash on these days (see Malachim 1, 8:65). But it is also because we rejoice with the kaparah that extends from Yom Kippur to 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, cleanses them from all aveiros.
There are several sources that we are pure from aveiros during these four days. Prominent among them is the pasuk (Vayikra 23:40) הראשון ביום לכם ולקחתם, "Take [the lulav] on the first day." The Midrash (Tanchuma Emor 22) explains that the 15th of Tishrei is called הראשון יום, the first day, because "It's the first day for reckoning aveiros..." Until then, we still have the kaparah of Yom Kippur.
The Satan doesn't have permission to prosecute or to lead us to sin on Yom Kippur. This is the one day a year that he is powerless. The Gemara (Yoma 20.) says that this is hinted in the word השטן, which is gematriya 364, one day less than the solar year. The Gemara learns from this that the Satan can only prosecute 364 days a year but is silent on Yom Kippur.
The Gemara discusses the word השטן, which is gematriyah 364, but שטן, without the heh as its prefix, is gematriya 359. Reb Yonoson 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.
These are hints to the purity from aveiros that we have on these days. These days are like an extension of Yom Kippur.
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 what about your Torah? 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 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 when an adult studies Torah during the four days between Yom Kippur and Succos, his Torah learning also sustains the world because on these days, he is clean from sin. It is even better than when a child studies Torah because an adult is a ועושה מצווה, obligated to study Torah (while a child isn't). The value of an obligatory mitzvah is greater than that of a mitzvah performed voluntarily. We can earn this merit when we study Torah on these days.
The Chasam Sofer zt'l was extremely happy between Yom Kippur and Succos because of the atonement we attained, to the extent that he couldn't focus on studying Torah. So, instead, 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 therefore, we should! And our studies at this time sustain the entire world.
It states (Yeshayah 55) בהמצאו 'ה דרשו, "Search for Hashem when He is near. The Gemara (Rosh Hashanah 18) asks, "When is Hashem nearby? 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 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 Hoshanah Rabba. These are the days that Hashem is nearby.
Yom Kippur is called Rosh Hashanah, and Hoshanah Rabba is called Yom Kippur. Between them, there are ten days. Regarding those days, it states בהמצאו 'ה דרשו, "Search Hashem when He is near."
