I'm not asking about the origin of this minhag. I'm inquiring of the purpose of the actual beating of the aravot on Hoshanna Rabba - i.e., the action, itself. What does the hitting of the aravot represent or do and what is its connection to Hoshanna Rabba specifically?
Rav Shimshon Raphael Hirsch writes in Chorev here (second paragraph) that after we have encircled the Bimah seven times we beat the Aravos as a means to remove any sense of Ga'avah (haughtiness / arrogance) from our hearts and the belief that our successes come from us alone. He says that the Aravah is symbolic of the 'chomer' lit. the material i.e. it is something that we receive from nature - it grows from the tree, wild and untreated, much like our natural abilities and talents. And with our human deficiencies, we wrongly assume that these talents are solely due to us. Therefore we specifically beat the Aravos to remind ourselves that everything is G-d given and beat out the natural arrogance that might express itself.
According to Rabbi Mansour, the purpose of beating the aravot against the ground is to beat the lips of the Satan such that he will be unable to testify against us in the coming year. The connection to Hoshanah Rabah is that on that day our final judgement is sealed. As he explains "The reason for this custom relates to the comment of the Midrash that the leaves of the Araba symbolize the lips. Specifically, the Arabot represent the lips of the Satan, and we bang them on the ground in order to silence the Satan so he does not prosecute against us. Hoshana Rabba is the day when our final judgment is sealed, and we therefore seek to silence the Satan in an effort to guarantee a favorable judgment. Furthermore, the Arizal (Rav Yishak Luria of Safed, 1534-1572) instructed that one should have in mind while banging the Arabot that the five beatings should correspond to the five letters in the Hebrew alphabet that appear only at the end of words (the final Kaf, the final Mem, the final Nun, the final Feh, and the final Sadi). These letters signify the Geburot – the harsh judgments in the world – and we bang the Arabot on the ground to express our desire to eliminate these judgments. The custom of the Arabot also commemorates the practice that was observed in the Bet Hamikdash to walk around the altar with an Araba on each day of Succot, and seven times on Hoshana Rabba. We commemorate this ritual by using Arabot on the day of Hoshana Rabba."