It states (Vayikra 23:42) בַּסֵּכֹּת תֵּשְׁבוּ שִׁבְעַת יָמִים "Live in succahs for seven days." The Gemara (Succah 2a) explains "אמרה תורה כל שבעת הימים צא מדירת קבע ושב בדירת עראי "For all seven days, leave your permanent home and dwell in a temporary abode." Why do we leave the strong walls of our home and live in a succah for seven days?
Menoras HaMaor (נר ג' כלל ד' ח"ו פ"א) explains, "The mitzvah of succah tells us that a person shouldn't trust in his tall and sturdy home, even if it is filled with a lot of good. He also shouldn't trust in the aid of a human, even if he is a powerful ruler of the land. One should trust in the One Who spoke and the world came into being... The mitzvah of succah teaches us this lesson. The mitzvah is performed after we finish gathering the crops of Eretz Yisrael, including the vineyard. This is a time of בעיטה [kicking, rebellion, when a person tends to rebel against Hashem because he feels he can manage on his own]. His treasures are filled with wealth; he has already returned from the field to his home in the city; he has put pitch on his roof, and he has closed all openings of his home, so the rain of the winter won't come in. Neither the wind nor any other elements will bother him. Therefore, Hashem commanded that one should leave his sturdy home and dwell in the succah. This will train him to trust in Hashem yisbarach, and to place upon his heart that all the good that he has received from the field was given to him with Hashem's will. He should remember that the protection for his life, and everything he has, is from Hashem yisbarach, and he shouldn't trust in the goods that he already has."
Sfas Emes (5645 ד"ה מצות) writes, "The mitzvah of succah is about bitachon in Hashem yisbarach, as Chazal say, צא מדירת קבע 'Go out from your permanent home'. [Go out of the mindset that you can rely on your wealth and your home.] Don't trust in your wealth or your property, only in Hashem."
Chazal (Succah 9b) say, העושה סוכתו תחת האילן כאילו עשאה בתוך הבית "If a person builds his succah under a tree, it is [pasul, just] as if he made the succah in his home." Some explain that there is a similarity between a succah built under a tree and a succah built in a home. A succah must be built under the heavens, without interference, so a person will know that he is dependent on Hashem, alone, Who dwells in heaven. He shouldn't trust in any person, and he shouldn't seek the protection of a home or a tree. He should rely on Hashem alone.
Some explain (quoting Reb Shamshon Refael Hirsh zt'l) that this is the reason מחובר when the schach that covers the succah is attached to the ground - is pasul. The schach must be detached from the earth. Generally, people are very "attached" and dependent on their assets and money. Therefore, we tell him, "For seven days, leave your permanent home, cleanse your mind from trusting in others, and dwell for seven days in a temporary abode, under the צילא דמיהמנותא shade of emunah. Train yourself to be "detached" from this world, and to trust in Hashem."
Chazal (Succah 28b) says, "When a child still needs his mother, he is exempt from sitting in the succah." The Gemara asks, "Who is a child who needs his mother? It is a child who, when he awakens from his sleep, calls Mother, Mother. The Gemara explains that if he calls "Mother" once, it is natural. Older children also do that when they wake up in the morning. But if he repeats "Mother! Mother!" the child is very young and very dependent on his mother, and is therefore patur from sitting in the succah. Some explain (אתה בחרתנו עמ' ק"ב) that the purpose of the succah is to train us to trust in Hashem. If the child calls "Mother! Mother!" he is trusting in human beings. He isn't prepared for the mitzvah and lesson of succah: to trust in Hashem alone.
Succos is called חג האסיף "the Yom Tov of gathering." This is because Succos is celebrated in the season when the new crops are brought into the home. The Sfas Emes (5645 מצות) ד"ה teaches that just as the new crops are the source of food for the entire year, so, too, brachos come down from heaven on Succos, which will be with us the entire year. Succos teaches us to trust in Hashem, and when we trust in Hashem, Hashem will bestow His blessings upon us.
The Kli Yakar (Vayikra 23:42) writes, "After people harvest their crops, they want to go home and live in a permanent dwelling. The Torah is concerned that this might lead to arrogance. Therefore, the Torah says כל האזרח whoever has plans to live in this world as a permanent resident, ישבו בסוכות, he must first dwell in the temporary succah. This will remind him to be humble and to know that he is a stranger in this world. He isn't a permanent resident here. With this realization, he won't place his trust in the walls of his home. He will trust in Hashem alone... This is what our forefathers did when they left Mitzrayim. They didn't live in high-class homes. They were surrounded by the seven clouds, where Hashem's honor was among them. They were בצל ש-ד-י in the shadow of Hashem, and not in the shadow of their homes, made from good wood."
The Gemara (Succah 2a) tells us that a succah higher than twenty amos is pasul. The Gemara explains that when the walls of the succah are so high, one will be sitting in the shade of the walls and won't receive shade from the schach. Therefore, the succah is pasul. (סיום למסכת סוכה, נדפס בסוף המסכתא) The Aruch I'Ner explains the problem. The walls of the succah represent this world. The schach represents Hashem's hashgachah pratis, Who gives him all his needs at every moment. "Schach higher than twenty amos is pasul because then he sits in the shadow generated from the walls and not from the schach. This represents a person who trusts in matters of this world to help him and is pasul, and not desired by Hashem." A person must trust in Hashem Who watches over him with hashgachah pratis and gives him all his needs, at every moment. Knowing this is the primary mitzvah of succah.
Shulchan Aruch (631:3) states, דֶּרֶךְ הַסִכּוּךְ לִהְיוֹת קַל, כְּדֵי שֶׁיֵּרָאוּ מִמֶּנּוּ הַכּוֹכָבִים הַגְדוֹלִים "It is the way of schach that one should be able to see through them the large stars." Pri Megadim (א"א תרל סק"ב) explains that the reason is so he can see the heaven, as it states (Yeshayah 40:26) שְׂאוּ מָרוֹם עֵינֵיכֶם וּרְאוּ מִי בָרָא אֵלֶּה "Lift up your eyes on high and see Who created these." He will understand that Hakadosh Baruch Hu created the entire world and leads it.