The Vilna Gaon famously pointed out that there are two mitzvot that one fulfills with his whole body – living in Israel and living in a sukka. Of course this can be a cute physical observation, that there is no specific part of the body to which the mitzva applies. However, it is more likely that the intention is that these mitzvot apply to one’s whole persona.
Others explain how the mitzva of sukka relates in a very deep way to the whole relationship between the fulfiller of the mitzva and his Maker. The Slonimer Rebbe (Netivot Shalom, Sukkot 3) refers to the Mahari Weil, cited by the Baer Heitev (639:1), who says that whoever fulfills the mitzva of sukka is like one who is a partner with Hashem in the creation of the world. Despite our love for the sukka, this seems surprising. Chazal have a similar statement – Whoever recites the Torah’s account of the "creation of Shabbat" is like one who is His partner in Creation (Shabbat 119b).
The Netivot Shalom explains that both Shabbat and Yom Tov are like reunions between the king, and his son who has been distanced from him. The difference is that on Shabbat, the son comes to "visit" the father, and on Yom Tov the father comes to visit his son. Of course, this fits as well or better regarding Sukkot, where the Zohar says that the sukka itself is infused with a special divine presence.
However, the Slonimer Rebbe takes it a step further in explaining the element of simcha that we find both in regard to Shabbat and to Sukkot. Regarding Sukkot, there is a very clear command to be joyful (Devarim 16:14). Regarding Shabbat, we find Chazal explaining the pasuk in regard to the use of the trumpet (Bamidbar 10:10- see Sifrei ad loc.) that "the day of your joy" refers to Shabbat. The commentaries have trouble with this, considering that there does not seem to be a mitzva of simcha on Shabbat.
The Slonimer Rebbe posits that the happiness is not referring to the way one is supposed to act on those days but to the essence of the days. Both Shabbat and Sukkot, he demonstrates, are like days of marriage between the Jewish People and Hashem. In fact the Avudraham says that different parts of Shabbat correspond to different stages in the creation of the nuptial union. The night is compared to the kiddishun (the betrothal); the morning is compared to the time of engagement during which presents are given; and the afternoon corresponds to the final stage of marriage when they start living together. Similarly, the sukka functions as a chupa (bridal canopy), so to speak, which makes Sukkot a time of internal happiness in addition to performing acts that create happiness.
As we who take part in these special days form a partnership of "marriage" with Hashem, we become partners in His world. In that way, it is as if one is a partner in the creation of the world, as he is a partner in everything that relates to Hashem.
May we feel the joy of the great relationship we forge with Hashem every Shabbat and, uniquely, on this holiday of Sukkot.