The Mishnah (Sukkah 4:1) lists seven different laws of Sukkot, one of whom is the law of aravot. The Mishnah (Sukkah 4:5) explains the law of aravot: In the time of the Bet ha-Mikdash the people would go to Motza to get aravot and then they would stand them by the side of the altar in order that the top of the aravot would be over the altar. Afterwards, the shofar would be blown, the people would circle the altar, and recite the phrase, anna hashem hoshiah na, anna hashem haslicha na, "G-d, save us, G-d, let us prosper." On the seventh day of Sukkot, the people circled seven times around the altar. The Mishnah did not discuss the reason for this ceremony.
The most likely explanation for the aravot ceremony is that it relates to prayers for rain since the Mishnah (Rosh Hashanah 1:2) records that Sukkot is the time of judgment for the water for the year. (This dating accords with the weather pattern in Israel that the rainy season begins in the fall after Sukkot.) Furthermore, in many societies, prayers for rain involved processions. (For one example, see The New York Times, July 6, 1908, "Italians pray for rain: Procession of 30,000 seeking to end disastrous drought in Apulia.) The increase in processions on the seventh day would then be because this was the final time to pray on Sukkot, and these processions on the seventh day were so important that they occurred even if the seventh day was on Shabbat (Mishnah Sukkah 4:6). Furthermore, the prayers "G-d, save us and G-d, let us prosper" can also relate to prayers of rain that the rain was needed for the people to live and to prosper. Finally, the blowing of the shofar in the ceremony of the aravot also connotes the idea of judgment.
Two questions remain. How did the placing of the aravot on the altar relate to prayers for rain? Also, on Sukkot there is the water libations ceremony which is also for rain (see Mishnah, Sukkah 4:9, and statement by R. Akiva in Talmud Rosh Hashanah 16a), why was there a need for two ceremonies (aravot and water libations) to pray for rain?
The answer is that the two ceremonies, aravot and water libations, worked together to form a joint symbolism. By the water libations, the altar was "being watered," and then by placing the aravot on the top of the altar, this made the altar green. Thus, at the end of the ceremony of the aravot the people would say how beautiful was the altar, end of Mishnah, Sukkah 4:5. In addition, if the people circled the altar holding aravot (the simple reading of the Mishnah Sukkah 4:5, but see Talmud, Sukkah 43b), then this also gave the entire area around the altar a green imagery. This symbolism of watering and greenery is the hoped for result from the prayers for rain, that there would be enough rain to make everything green that the crops would grow.