Shemini Atzeres and Simchas Torah
The Greatness of the Day
Complete Achdus With Hashem:
The Me'or Einayim (Parshas Haazinu) writes: "All the avodos that we do during this month – Rosh Hashanah, Yom Kippur, shofar, sukkah, and lulav – the purpose of it all is because of Shemini Atzeres, when we unite with Hashem with complete unity."
The World Stands on the Merit of These Days:
Sefer Raza D'Uvda (Leil Bais D'Sukkos) quotes my ancestor, Rav Eliezer Zev of Kretchnif zy"a, as saying in the name of his grandfather, Rav Uren Leib of Premishlan zy"a, that the entire world stands upon the three days of Hoshanah Rabbah, Shemini Atzeres, and Simchas Torah.
No Need for a Specific Mitzvah:
The Chasam Sofer zt"l (Drashon L'Sukkos, page 54) writes that every holiday has a special mitzvah, such as eating matzoh on Pesach, the Shtei Halechem on Shavuos, shofar on Rosh Hashanah, fasting on Yom Kippur, and the Daled Minim and sukkah on Sukkos. Shemini Atzeres, however, has no special mitzvah. Since it is so holy in and of itself, it doesn't need any mitzvah besides "v'hayisa ach sameach" (you shall only be happy). On this day, we don't need anything else for our avodas Hashem.
Entering the King's Chamber:
Sefer She'eris Yisroel (Shaar Hazemanim) relates that one year, after the hakafos on Simchas Torah, the Rebbe of Vilednik zy"a related a moshol of a powerful king who designated one day in the year when any subject could enter his castle to see him in his glory. He knew that the regular guards wouldn't allow the lowliest people to enter, so he assigned a special guard to be in charge on that day. He took the man who usually guarded the horses and gave him this job. Since this man was lowly himself, he happily allowed all the lowliest people to enter. And since he was opening the door for all the simple people, he too was able to see the face of the king.
The Rebbe then said in his humility, "I am like that guard of the horses. I am lowly person. But since I help people see Hashem's glory on this day, I also am able to see it."
To be Able to Get Home:
Sefer Imrei Kodesh Hashalem writes that Rav Yissochor Dov of Belz zy"a quoted Rav Uri of Strelisk zy"a as saying: "After Hashem helps a Jew fulfill the mitzvos of sukkah, and he merits to feel the holiness of the walls and the schach and perform all the associated mitzvos, how is it possible for him to return to his regular home? It is a great kindness of Hashem that He gives us the holiness of Shemini Atzeres right after Sukkos, which helps us transition back to our homes. Without this, it would be impossible to go straight from such holiness to our regular lives."
Tov Li Toras Picha:
Sefer Darchei Hayashar V'Hatov relates that one year, Shemini Atzeres fell on Shabbos and the crowd at the Liska Rebbe zy"a's tish was grabbing the kugel. The Rebbe stood up and declared, "I can give you something even better than kugel."
He then encouraged the working men to financially support the Torah scholars, and he told the talmidei chochomim how to interact with the balhabatim. He concluded by saying, "Nu, didn't I give you something better than kugel?"