"Sound the shofar on the new moon, in concealment to the day of our festival" (Psalms 81:4). In this cryptic verse lies the deeper significance of Sukkot and its sister-festival, Shemini Atzeret/Simchat Torah.
In the Jewish calendar, the month follows the phases of the moon. The "new moon" -- the point at which the moon emerges from its monthly concealment -- marks the beginning of a new month. The apex of the month is the 15th -- the night of the full moon, when the moon attains the high point of its potential to reflect the sun's light and illuminate the earth.
The month of Tishrei is the most spirituality-rich month in the Jewish year. The festivals and special days-- Rosh Hashanah, the Ten Days of Repentance, Shabbat Shuvah, Yom Kippur, Sukkot, Shemini Atzeret/Simchat Torah, Shabbat Bereishit -- follow in succession, with hardly any "ordinary days" in between. It's a time to fuel up on the spiritual resources -- awe, teshuvah, connection, singularity, joy, unity, wisdom, commitment -- that will drive our lives for the rest of the year.
The first of Tishrei is Rosh Hashanah, which opens the "Days of Awe" that characterize the first part of the month, culminating in Yom Kippur on Tishrei 10th. Then the mood and texture of Tishrei shifts dramatically to the "Season of Our Rejoicing" that begins with the festival of Sukkot on the 15th and continues through Shemini Atzeret and Simchat Torah. The half of Tishrei is marked by solemnity, the second by elation; but the Chassidic masters explain that these are simply the "hidden" and "revealed" dimensions of the same elements.
The essence of Rosh Hashanah is our crowning of G-d as our "king". A coronation, explain the Chassidic masters, is effected by two things -- unity and joy: a people joyously unite to select, accept and submit an exalted figure who embodies their collective identity and innermost strivings (if the coronation lacks either joy or unity, chassidic teaching explains, it results not in a true king, but merely in a "ruler"). But there is also a third element without which the coronation could not happen -- awe. And the nature of awe is that it eclipses and mutes the joy. Sukkot, then, is simply the revelation of Rosh Hashanah. The joy and unity that are the essence of our commitment to G-d, and which were "concealed" by the awe that characterizes the first days of Tishrei, erupt on the 15th of the month in the form of the festival of Sukkot.
In the words of the Psalmist, "Sound the shofar on the new moon, in concealment to the day of our festival." Our crowning G-d king with the sounding of the shofar on the 1st of Tishrei ("the new moon") remains in concealment until "the day of our festival", the full moon of Sukkot, when it manifests itself in a seven-day feast of joy.
And what Sukkot is to Rosh Hashanah, Simchat Torah is to Yom Kippur. The essence of Yom Kippur is that it is the day we received the Second Tablets, completing the Giving of the Torah (which began on Shavuot) and bringing into our lives the Torah's ultimate essence -- teshuvah. There is nothing more liberating and exilarating than teshuvah -- the power to "return" to the quintessential core of one's being, transcending time, space, habit and "character", transcending all that circumscribes the soul's truest self and truest strivings. But again, the essence of Yom Kippur is submerged and concealed within the solemnity that accompanies the business of teshuvah doing -- fasting, regretting and confessing our wrongdoings, resolving not to repeat them, praying for forgiveness. It is only on Simchat Torah that the joyous essence of teshuvah is manifestly celebrated.
For more on Sukkot and Simchat Torah, and the special mitzvot and observances -- the sukkah, the "four kinds", the hakafot, etc. -- that are the tools and implements of their unity and joy, see the following essays and stories: