The Sages teach that the month of Nissan when the Exodus took place is the beginning of the spring and the weather begins to be very temperate - in fact the weather is so good that it was a time that everyone would move out of their stuffy homes and live in huts (the equivalent of todays' bungalows, I guess!). The Sages explain that Sukkot was deliberately moved by God to a less pleasant time of the year to show the Jewish Nation's allegiance to God - that we move out to the Sukkah even though the weather is already beginning to change for the worse.
However, there is also a deeper more esoteric concept behind the date as well. The Talmud states, Tractate Sukkah 11b, that Rabbi Eliezer said the Sukkot were the clouds of glory and Rabbi Akivah said they were booths. Rashi cites Rabbi Akivah's view and is the view that appears in the Shulchan Aruch. There is, however, an incredible insight from the Vilna Gaon to reinforce Rabbi Eliezer's view. The Vilna Gaon writes that when the Jewish People made the Golden Calf the Clouds of Glory disappeared. Moshe descended from Mount Sinai on Yom Kippur with the news that they had been forgiven for building the Golden Calf. On the next day, the eleventh of Tishrei, Moshe commanded the Jewish People to start the process that would build the Mishkan. The people brought their contributions on the twelfth and the thirteenth and on the fourteenth Moshe had to command everyone to stop bringing donations as they had more than enough. On the fifteenth of Tishrei they began building the Mishkan and the Clouds of Glory returned. Says the Vilna Gaon that is one of the reasons why we celebrate Sukkot on the fifteenth of Tishrei because the Sukkot are like the Clouds of Glory!
