The Mishna in R"H writes that unlike hallel which is said right after shacharis, tekiyas shofar is delayed until musaf. The gemara explains that this is a takanah from times when there was a danger of the enemy misinterpreting the gathering of Jews as a rebellion and shofar blast as a call to arms. Once shacharis passed and it was apparent that this was just a religious service, there was no danger anymore. If so, asks the Yerushalmi (perek 4 halacha 8), why was their not the same concern with respect to hallel, i.e. that the noise of the singing of hallel would be misinterpreted as an uprising? The gemara answers (compare with the Bavli) that not everyone is in shul that early (a makor for coming to shul late on Y"T : ) and a small minor gathering would not be misinterpreted as a rebellion. However, says the Yerushalmi, everyone is in shul for tekiyas shofar (amazing -- it's still true to this very day.) The Yerushalmi ends off with a derasha: "Amar Rav Yonah, ksiv 'V'osi yom yom yidroshun' -- zu tekiya v'aravah." Explains the Pnei Moshe, on the day of Rosh haShana by tekiyos and the day of Hoshana Rabbah by aravah everyone is in shul to daven together.
Why these two days in particular? Why are davka tekiyas shofar and the mitzvah of aravah times of 'V'osi yom yom yidroshun?'
Achronim explain that aside from being a kiyum of mitzvas shofar, the tekiyos which we blow during shmoneh esrei of Rosh HaShana are also a kiyum of tefilah. Sometimes a person cannot even get the words out to daven -- they can just sigh or cry and that's it. That's the tefilah of tekiyas shofar -- a tefilah without words. Rosh haShana is an opportunity for our prayers to be heard even if we can't articulate the words.
The four minim represent four types of Jews. The esrog which has a nice taste and nice smell represents the tzadik who as Torah and mitzvos under his belt. At the other extreme is the aravah, that has neither taste nor smell and is bereft of ruchniyus. What zechus does the aravah-Jew have to call upon his his tefilos? None. The shape of the aravah resembles closed lips, says the Sefas Emes. Yet on Hoshana Rabbah, even those closed lips that have no zechuyos to call on can have their prayers answered (see this post as well).
Rosh haShana starts the season of the y'mei ha'din; Hoshana Rabbah ends it. The bookends are days of tefilah, days when Hashem is there to listen, whether you can get the words or not, whether you come with zechuyos or come with empty hands and just beg to be heard.