why IS sukkot right after yom kippur?

(33) The LORD spoke to Moses, saying: (34) Say to the Israelite people: On the fifteenth day of this seventh month there shall be the Feast of Booths to the LORD, [to last] seven days. (35) The first day shall be a sacred occasion: you shall not work at your occupations; (36) seven days you shall bring offerings by fire to the LORD. On the eighth day you shall observe a sacred occasion and bring an offering by fire to the LORD; it is a solemn gathering: you shall not work at your occupations.

This commandment is a parable that describes two people who contend in a legal battle before the King. Although the two people were not informed in whose favor the judgment was issued, they can figure it out by looking to see who walks out still holding his weapons. Thus Israel along with the nations of the world face the Divine judgment on the Day of Atonement. We cannot tell which one emerged victorious, which is why God commanded You shall take for yourselves ... You can see from the fact that Israel emerges from before God holding its palm leaves and citrons, its lulav and etrog, that Israel emerged victorious.​​​​​​​

For it has been taught: "For I made the children of Israel to dwell in booths [sukkot]" (Leviticus 23:43): These were clouds of glory, the words of R. Eliezer. R. Akiva says: They made for themselves real sukkot.

Help us, Lord, to lie down in peace, and awaken us again, our Sovereign, to life. Spread over us Your shelter of peace; guide us with Your good counsel. Save us because of Your mercy. Shield us from enemies and plague, from starvation, sword, and sorrow. Remove the evil forces that surround us. Shelter us in the shadow of Your wings, O G-d, who watches over us and delivers us, our gracious and merciful Ruler. Guard our coming and our going; grant us life and peace, now and always. Spread over us the shelter of Your peace. Praised are you, Lord, who spreads the shelter of peace over us, over all G-d's people Israel, and over Jerusalem.

(42) You shall live in booths seven days; all citizens in Israel shall live in booths, (43) in order that future generations may know that I made the Israelite people live in booths when I brought them out of the land of Egypt: I the LORD your God.

(1) A sukkah taller than twenty cubits is invalid. Rabbi Yehudah validates it. And one which is not ten hand-breadths tall, or does not contain three walls, or whose whose [area of] sun is greater than its shade is invalid. Regarding an old sukkah, Beit Shammai invalidate it, and Beit Hillel validate it. What is an old sukkah? Any [sukkah] which one made it thirty days before the festival. But if one made it for the sake of the festival, even [if he made it] from the beginning of the year, it is valid.

Rava replied: From the following verse, "You shall live in booths for seven days" (Leviticus 23:42) the Torah declared: For seven days leave your permanent dwelling place and dwell in a temporary dwelling place. Up to twenty cubits [high] one makes his home a temporary one; higher than twenty cubits, one does not make his home temporary, but permanent. Abaye said to him: If so, if he made walls of iron and placed the [proper] skhakh over them, would it also not be a valid Sukkah. He answered him: This is what I mean to tell you: Up to twenty cubits, which a man makes his temporary home, even if he makes it permanent,he has fulfilled his mitzvah; [but in one] higher than twenty cubits, such as a man makes his permanent home, even if he makes it temporary, he has not fulfilled his obligation.

(1) The pipes [were played sometimes on] five [days], and [sometimes on] six days. This means, the pipes [music] played on during the water-drawing, which does not supersede either the Sabbath or the festival. They [the sages] said, "Any one who has not witnessed the rejoicings at the water-drawing, has never witnessed real rejoicing in their entire lives." (2) At the expiration of the first holy day of the festival they descended into the women's court, where great preparations were made [for the rejoicing]. Four golden candelabras were [placed] there, with four golden basins to each; and four ladders [were put] to each candelabra, [on which ladders stood] four lads from the rising youth of the priesthood, holding jars of oil, containing 120 lugs, with which they replenished [fed] the basins. (3) The cast-off breeches and belts of the priests were torn into shreds for wicks, which they lighted. There was not a court in Jerusalem that was not illuminated by the lights of the water-drawing. (4) Pious and distinguished men danced before the people with lighted flambeaux in their hands, and sang hymns and lauds before them;, and the Levites accompanied them with harps, psalteries, cymbals, and numberless musical instruments. On the fifteen steps which led into the women's court, corresponding with the fifteen songs of degrees, stood the Levites, with their musical instruments and sang. At the upper gate, which leads down from the court of the Israelites to the court of the women, stood two priests, with trumpets in their hands. When the cock [first] crowed they blew a blast, a long note and a blast. This they repeated when they reached the tenth step, and again [the third time] when they got into the court. They went on, blowing [their trumpets] as they went, until they reached the gate that leads out to the east. When they reached the gate that leads out to the east, they turned westward [with their faces towards the Temple], and said, "Our ancestors, who were in this place, turned their backs on the Temple of the Lord, and their faces towards the east; for they worshipped the Sun towards the east: but we lift our eyes to God." R. Jehudah says, they repeated again and again, "We belong to God, and raise our eyes to God."