The Biblical origin is in the Book of Vayikra; specifically, in Parshat Emor, where the Torah commands "And you shall take for yourselves on the First Day the fruit of a beautiful tree, the branches of date palms, branches of the myrtle tree, and branches of the willow tree, and you shall rejoice before Hashem, your G-d, for Seven Days." (Vayikra 23:40)
The Commandment is to take these four species together as a unit, and to shake them together in all directions, at various times on Sukkot.
One possible explanation is that we are taking these four elements from nature, and demonstrating that Hashem rules over nature everywhere and, by fulfilling this Command throughout the generations of our People, also at all times.
Two additional aspects of this "Group of Four" are as follows:
Holding these four in a tight bond represents the unity that is Hashem's goal for the Jewish People. The bond represents the conversion of a set of separate individuals into a People, which is far greater than any individual in both the Crown of Torah and the Crown of Good Deeds, and is far more deserving than any individual of the blessings of Hashem.
For each of the Four Species, the Mishnah in Masechet Sukkah compares the stolen article to its dried out and lifeless form, which is absolutely invalid. This is derived from a word in the Torah, and is also very understandable. It is derived from the word "yourselves" in the expression "And you shall take for yourselves the fruit of a beautiful tree, ," implying that one's ownership is required."
But the explanation is to be found in the Talmud. The reason that the stolen "Lulav," for example, cannot be used as part of the fulfillment of a Divine Command, is that it would then be a "Command performed By Means of a Sin," which is self-contradictory! It is obvious that an Act would not be pleasing to Hashem, if it comes at the price of violation of one of His Own Commands.