One of the names of the Shabbos between Rosh HaShanah and Yom Kippur is Shabbos Teshuvah, the "Shabbos of Repentance," since it is the Shabbos of the Aseres Yemei Tes, the "Ten Days of Repentance," that begin with Rosh HaShanah and conclude with Yom Kippur.
Since all ten days share the same name, it follows that they all constitute a single entity whose beginning is Rosh HaShanah and whose conclusion is Yom Kippur. It is therefore easy to see that preparation for Yom Kippur, the culmination of the "Ten Days of Repentance," begins at Rosh HaShanah.
The main commandment of Rosh HaShanah is to sound the shofar. The singular point of Yom Kippur is the service of the Kohen Gadol, the High Priest; during the rest of the year the service in the Temple could be conducted by other priests, but on Yom Kippur the entire service had to be performed by the Kohen Gadol.
Wherein lies the connection between Rosh HaShanah, the beginning of the Ten Days of Repentance, and its conclusion on Yom Kippur? Seemingly, the service of sounding the shofar and the service of the Kohen Gadol on Yom Kippur are entirely different.
There were two aspects to the Kohen Gadol's service on Yom Kippur: the service that he performed in garments of gold outside the Holy of Holies, and the main aspect of his service, performed within the Holy of Holies while wearing simple linen garments.
When the priests would perform the service in the Temple they would wear garments that served both to "honor and adorn." The reason is that one should employ that which is most beautiful in the service of G-d. Truly, this was also why the Kohen Gadol performed the service outside the Holy of Holies in garments of gold, since these are most impressive.
This being the case, why was it that within the Holy of Holies, where only the Kohen Gadol was permitted, and only then on Yom Kippur, at which time he achieved atonement for the entire Jewish people, that he performed in simple linen garments?
Although the destruction of the Temple led to the cessation of the Kohen Gadol's service there on Yom Kippur, the spiritual Temple found within each and every Jewish soul was never destroyed.
Within this spiritual Temple there also exist different degrees of service according to the specific time of year. With the arrival of Yom Kippur, each Jew - serving as the Kohen Gadol of his own Temple - is to perform the service of the Kohen Gadol.
This service is composed of serving outside the Holy of Holies in garments of gold, while inside the Holy of Holies of his soul he is to serve in linen garments:
This means that one should adorn one's service when one is serving outside the Holy of Holies so that those physical things that are most dear to the person ("gold") are used for spiritual ("Temple") purposes.
However, when it comes to the "Yom Kippur" and the "Holy of Holies" of a Jew's soul, physical adornment is of no benefit, for the operative word is simplicity - a service of simple faith and self-sacrifice wherein all Jews can serve equally.
This theme of simplicity is also found with regard to the sounding of the shofar, a simple horn of an animal that does not emit melodious sounds, but "merely" gives forth simple notes.
On Rosh HaShanah a Jew comes before G-d with a simple heartrending cry, the plaintive sound of the shofar that comes from the depth of his heart, beseeching G-d for a good and sweet year.
Through this service during the Ten Days of Repentance, culminating in his simplicity on Yom Kippur, he merits to have his request granted - he is "sealed" for a good and sweet year.
Based on Likkutei Sichos, Vol. II, pp. 411-413
Nitzavim