What is the purpose of leaving our homes and entering the Sukkah on Yom Tov? What is the purpose of the Lulav on Sukkot, and what does the Lulav truly represent? Rava states in the Gemara Sukkah (דף לב.), that a Lulav whose leaves emerge from only one side is called a Ba'al Mum, meaning that it is considered to have a blemish. Since the term Ba'al Mum is usually associated with Korbanot, Lulav seems to be a type of Korban. What does it mean that a Lulav is a type of Korban?
Before we answer the question, we first have to answer the question of why we begin reciting "Mashiv HaRuach..." on Shimini Atzeret One of the purposes of the four Minim is to plead to Hashem for water. This is mentioned by the Gemara in Taanit, where the gemara says (דף ב:) "just like none of the four Minim can live without water, so too we can't live without water." Therefore, the Lulav itself has a similar purpose to Korbanot, which are generally used to beseech Hashem.
There is an indication in the Torah that the Lulav is a Korban. We see by the discussion of the Yamim Tovim in ויקרא פרק כג, that there is a special Korban mentioned for each of the holidays, yet regarding Sukkot there is no mention of a Korban. The only things that are mentioned are the Mitzvot of the four Minim and the Sukkah. One may conclude that the Torah is implying that on some level, the Lulav, with the other Minim, is a type of Korban.
Furthermore, the Gemara in Sukkah (דף לז:) compares the Lulav and the Korban Shtei HaLechem (brought on Shavout). The person who brings the Korban does נענעוים, wavings in six different directions (up, down, right, left, forward, and backward). Rabbi Yochanan says that one does this to show his recognition that Hashem has the power to sustain the world. Similarly, one also does נענעוים to the Lulav to "turn away bad winds," to show and recognize that Hashem is the only Being capable of influencing "bad winds." Waving the Korban Shtei HaLechem parallels waving the Lulav, both symbolizing that Hashem alone has the power to sustain the world.
The Rambam (הלכות לולב פרק ח, יב - טו) says that on Sukkot there is a unique Simcha in the Bet HaMikdash, which we do not have on other holidays. Why is Sukkot singled out? Sukkot is different because it is, spiritually speaking, the most dangerous time of the year. This time of year is the harvesting season, therefore, it is easy for the farmer to think that he made all the produce, because he is the one who worked so hard turning the soil and planting the seeds. Since this time period is fraught with spiritual danger, it is imperative that we recognize that Hashem is the one who sustains the world and makes our plants grow. One way of recognizing this is by saying "Mashiv HaRuach..." which demonstrates that we recognize that Hashem is the one controls the amount of water that we receive during the year.
There is another way of showing that Hashem controls everything. Chazal say (שם דף יא:) that the Sukkah commemorates theענני הכבוד , the Divine clouds which protected Bnai Yisrael in the Midbar. When Bnai Yisrael were in the desert and they saw the Hashem's cloud every day, it was impossible to deny that everything came from Hashem. However, once Bnai Yisrael entered Eretz Yisrael and no longer had the ענני הכבוד above them, it became possible for them to say that their success was due to their efforts and not Hashem's. This is why we move out of our homes on Sukkot, to remind us of the ענני הכבוד, and to reinforce the fact that everything we have is because of Hashem.
This is the reason why there is a special Simcha in the Bet HaMikdash on Sukkot. It is because the Bet HaMikdash is a place where it is as easy to recognize Hashem as it was in the desert. Each of our Sukkot can also be viewed as a mini Bet HaMikdash where we can clearly recognize Hashem. There is still one question which we have not answered, and that is what does the Lulav represent? The Lulav is a type of Korban which is performed on Sukkot in your mini Bet HaMikdash (shul or Sukkah, depending on one's custom). The purpose of the Lulav is to show that Hashem is the one who sustains the world and decides its fate. It also acts as an appeasement to Hashem, which tells Him "Hashem, you are the only one who can sustain the world, and it is because of you that my produce grew."
(I would like to thank my Rebbe from NCSY Kollel, Mark Smilowitz for teaching me the material in this article.)