Don't Say Ata Haraysa With the Yeitzer Hara:
It was the first year after the wedding of Rav Levi Yitzchok of Berditchov zy"a. He was living in the home of his father-in-law, and his greatness was not yet known to the world. That year, he was given the honor of standing before the aron and reciting the pesukim of Ata Hareisa.
He walked towards the front of the shul and picked up the talis. However, he then put it back down and stood motionless for a few minutes. He then picked up the talis again, and immediately put it down again. He did this several times, while everyone was waiting for him to start Ata Hareisa. Finally, his voice called out, "Do you think you're a chasid and a lamdan? Just start Ata Hareisa already!"
He then turned around and went back to his seat.
Everyone thought he was acting very strangely. His father-in-law, who was a wealthy, respected man, was very embarrassed. Still and all, he didn't want to question his son-in-law. After the Hakafos, however, he couldn't hold himself back, and he asked Rav Levi Yitzchok to explain himself.
Rav Levi Yitzchok said, "When I was about to put on the talis to start Ata Hareisa, the yeitzer hara came to me and said that it wanted to say the pesukim along with me. I asked it: Who are you? Why should you get to say Ata Hareisa with me? It replied: And who exactly do you think you are? I answered: I am a lamdan. It then said: I am also a lamdan.
"I asked it: I learned Shas and Poskim by well-known gaonim. Who did you learn from? It answered: I was with you there. So I learned by the same gaonim.
"I then said: But I am also a chasid. It said: I too am a chasid. I told it: I learned chasidus by famous tzadikim. Who taught you chasidus? It answered: I was with you there and learned the same things you did.
"When I saw that I had no answers that would get the yeitzer hara to leave, I put down the talis and said: If you're such a lamdan and chasid, then you should say Ata Hareisa by yourself."
Similarly, it is written in Sefer Imros Moshe that Rav Moshe of Kobrin zy"a related that Rav Moshe Iver zy"a, (one of the Gedolim of Vilna) was once honored with the reciting of Ata Haraisa on Simchas Torah, as it is customary to grant this honor to a distinguished member of a congregation. When the gabbai announced that he was receiving this honor, he called him up using the more distinguished word "eich". He said, "Men iz eich mechubad mit Ata Haraisah." (In Yiddish, an individual can be called by the more personal "du", you, which is a singular tense, or by the more honorable title "eich", you, which is technically the plural tense.)
Rav Moshe responded, "Why are you using the plural word 'eich?'... Certainly, your intention is that both my yeitzer hara and I are receiving this honor. My yeitzer hara tells me that he is always with me. He says, 'We learn together. We daven together.' That must be why you are including him in the Ata Haraisa together with me."
He then put his talis down on the table and said, "If that is the case, let him say it by himself. Yeitzer hara, you go ahead and say Ata Haraisa alone. I don't want to say it together with you."
Tears of Blood:
The Rashab of Lubavitch zy"a asked his father, the Maharash of Lubavitch zy"a to explain the concept of Hakafos to him. He answered that the word "Hakafos" connotes beseeching Hashem with tears of blood, as is stated: "My tears were my bread." Thus, we dance with the sifrei Torah with great joy on the outside, but on the inside, we are weeping tears of blood.
The Joy of the Angels:
Sefer B'Yashishim Chochmah (page 324) relates that a group of chasidim once went to take leave of the Tzvi L'Tzadik of Bluzhov zy"a after the yomtov of Sukkos. He asked them, "Why haven't you asked me why most chasidim in Chutz La'aretz make Hakafos only by night on Shemini Atzeres and not by day, but I make Hakafos even by day?"
The Rebbe continued to tell them a story: Rav Moshe Sofer of Pshevorsk zy"a would travel every year to spend Shemini Atzeres with his Rebbe, Rav Elimelech of Lizhensk zy"a. One year, his wife gave birth to a baby before Yom Kippur. He needed to be home for the bris, so he didn't go to Lizhensk that year. On Shemini Atzeres, he was filled with longing for his Rebbe. So what did he do? He went down to his basement, where he had a barrel full of whiskey. He drank a little bit, and danced around the barrel, and continued to do this all day.
On the day after Simchas Torah, he traveled to Lizhensk. Rav Elimelech was by the door waiting for him, because he saw with ruach hakodesh that he was coming. He told him, "Tell me the truth. What did you do on Shemini Atzeres? I saw that the malachim in Shomayim were smiling with joy because of you."
Rav Moshe told him that he had danced around a barrel.
The Bluzhover concluded, "Hakafos like these create great joy in Heaven, which is why we make them in the morning as well."
Canceling the Satan Through Hakafos:
The Rashab of Lubavitch zy"a (quoted in Sefer Toras Shalom) said in the name of his father, the Maharash of Lubavitch zy"a, that on Simchas Torah, there is no Satan and no angel can harm Klal Yisroel. On this day, all the Divine angels hold back the Satan and tell it, "Look at how much this nation loves Hashem. They come with their entire families to the bais medrash to rejoice with His Torah. How could you speak badly about them?"
They laud the nation for kissing and dancing with the sifrei Torah, and this causes a great joy in Heaven. The Satan is so ashamed to say anything bad about them that it goes into hiding for the day.