On Shabbos chol hamoed Sukkos we read from Parshas Ki Sisa. The simple reason this portion was chosen is because it mentions the shalosh regalim. But the kriyas haTorah also discusses the second luchos, the thirteen attributes of divine mercy, and the dialogue between Hashem and Moshe Rabbeinu about whether a malach or Hashem himself will lead Klal Yisrael. What do these other topics have to do with the yom tov of Sukkos?
The Tur (Orach Chaim, 625) asks a famous question. If we sit in sukkos to remind ourselves of the clouds of glory that surrounded Klal Yisrael when they left Mitzrayim, then why do we celebrate the yom tov of Sukkos in the month of Tishrei? Since Klal Yisrael left Mitzrayim during Nissan, we should celebrate Sukkos then.
The Vilna Gaon (Shir Hashirim, 1:4) answers that on Sukkos we are not commemorating the clouds of glory that enveloped Klal Yisrael when they first left Mitzrayim because those clouds disappeared after cheit ha'eigel. But after Hashem forgave Klal Yisrael for cheit ha'eigel on Yom Kippur, they received the command to build the Mishkan, and when they started building the Mishkan on the fifteenth of Tishrei the clouds of glory returned. It is this return of the clouds of glory that we commemorate by sitting in sukkos because this showed that Hashem had forgiven Klal Yisrael for cheit ha'eigel and He was willing to rest His Shechina on them once again. Since the clouds returned during the month of Tishrei, we celebrate Sukkos specifically at this time.
This idea can help explain why we read Parshas Ki Sisa on Shabbos chol hamoed Sukkos. At the beginning of the kriyah, Moshe Rabbeinu asks Hashem not to rest His Shechina on any nation except Klal Yisrael, "V'niflinu ani v'amcha mikol ha'am asher al pnei ha'adama - and I and your people will be distinct from all other nations of the world. (Ki Sisa, 33:16). Later, after Hashem teaches Moshe the thirteen attributes of mercy, Moshe once again asks, "Yeilech na Hashem b'kirbeinu - let Hashem dwell in our midst" (34:9), and Hashem responds, "I will establish a covenant (with you); I will make distinctions (e'eseh niflaos) with your people." Rashi interprets that here Hashem finally agrees to Moshe's request that Klal Yisrael should be different (v'niflinu) than all other nations in that Hashem should rest His Shechina only on them.
These pesukim are especially appropriate for the yom tov of Sukkos because, as the Vilna Gaon explains, it was on Yom Kippur that Hashem forgave Klal Yisrael and agreed to rest His Shechina on them, and then on Sukkos He reaffirmed that commitment by returning the clouds of glory which represent the Shechina.
The return of the clouds of glory on Sukkos could be another reason why we refer to Sukkos as zman simchaseinu - the time of our happiness. The simple explanation is that this refers to the happiness of the farmer who rejoices in the new harvest. Others explain that Sukkos is a time when we feel simcha for the atonement we achieved on Yom Kippur. But in light of the Vilna Gaon's comment, it would seem that Sukkos could be called zman simchaseinu because the fifteenth of Tishrei is the time that Klal Yisrael saw Hashem's intense love for them when He returned the clouds of glory and He established a covenant to rest His Shechina only on them. This realization that we enjoy such a special relationship with Hakadosh Boruch Hu is a source of great simcha and it transforms Sukkos into zman simchaseinu.
The expression of love between Hakadosh Boruch Hu and Klal Yisrael reaches a climax on Shemini Atzeres and Simchas Torah. On Sukkos we sacrifice a total of seventy bulls but on Shemini Atzeres we sacrifice only one bull. Rashi (Pinchas, 29:35-36) quotes the Midrash which explains that the seventy bulls we sacrifice on Sukkos correspond to the seventy nations of the world, while the one bull we sacrifice on Shemini Atzeres represents Klal Yisrael. Why do we bring korbanos corresponding to the nations of the world only on Sukkos? The answer is that the korbanos of Sukkos and Shemini Atzeres demonstrate the unique bond between Hashem and Klal Yisrael. The reality is that Hashem sustains the entire world and, in truth, the yom tov of Sukkos which celebrates the new harvest is relevant to all of the nations. But Hashem has a special relationship with Klal Yisrael that is expressed by the private feast that Klal Yisrael enjoys with Him on Shemini Atzeres (see Rashi there).
Shemini Atzeres is the culmination of the yom tov of Sukkos because it serves as a contrast to the universal yom tov of Sukkos. Shemini Atzeres is Hashem's expression of love for Klal Yisrael. It shows the unique relationship that Klal Yisrael has with Him. There are no special mitzvos on Shemini Atzeres because the simcha of Shemini Atzeres comes from simply feeling a sense of closeness to Hakadosh Boruch Hu. On Shemini Atzeres we don't need a simcha shel mitzvah to help us express our joy because we are overcome by a feeling of "nagilah v'nismecha bach - we will rejoice and be happy with You" (Shir Hashirim, 1:4).
And that is why Simchas Torah is linked to Shemini Atzeres. When Hakadosh Boruch Hu forgave Klal Yisrael on Yom Kippur he gave them the second luchos, the gift of Torah, to demonstrate his love for them. On Shemini Atzeres this love is expressed once again when Hakadosh Boruch Hu asks of Klal Yisrael to be his special guest for one last day of yom tov. We reciprocate by rejoicing with the Torah and showing how much we appreciate his gift and his expressions of love for us.