Granting all Desires:
The Zohar Hakadosh (Chelek Gimmel, 32A) states: "At this joyous time (of Shemini Atzeres), no entity is found with the King except for Yisroel alone. And when one sits alone with the King, he can ask for whatever he wants."
Rav Pinchos of Koretz zy"a (Imrei Pinchos, Shabbos U'Moadim, Ois 201) states that this only is true if we "sit with the King", meaning that we focus on our connection to Hashem and don't think about any mundane matters.
Geshem-Tal:
Sefer Siach Zekeinim (Chelek Gimmel, page 73) relates that one year while Tefilas Gashem was being recited on Shemini Atzeres, the Ateres Yeshua of Zhikov zy"a stood next to the amud and was silent for a long time. He then turned to the mispalelim with a fiery face and said, "The Gemara says (Gittin 2A) that when a witness brings a get (divorce document) from overseas, he must say that it was written and signed in front of him. 'Get' is an acronym for 'geshem tal.' The Gemara says (Taanis 2A) that Hashem does not give three keys to any agents. They always remain in His hand. One of these is the key to rain. Thus, Hashem is the one who brings the 'get' of geshem and tal overseas. Therefore, He must, so to speak, say that it was written for a good year and signed for a good year."
He then began to sing Yisgadel V'Yiskadesh Shmei Rabbah.
Morid Hageshem:
Rav Yankele of Pshevorsk zy"a (Yud Gimmel Oroso, Chelek Gimmel, page 279) quotes Rav Hersh of Rimanov zy"a as explaining the reason we start to pray for rain on Shemini Atzeres by saying that we have gone through the month of Elul, Selichos, Rosh Hashanah, Yom Kippur, and Sukkos, which were all days of pure ruchnius. We now say to Hashem: "Mashiv haruach. Let the ruchnius sit in its place for now. U'Morid Hagashem. And send down a lot of gashmius."