The Midrash states, "When the Jewish people take the shofar and blow it before Hakadosh Baruch Hu, Hashem stands up from the throne of judgment and sits on the throne of compassion."
The Ahavas Shalom zt'l explains how this happens:
On Rosh Hashanah, Hashem finds Himself sitting on the throne of judgment to judge the Jewish people, and Hashem asks Himself, "How did I get here? Why should I judge the Jewish people whom I love?" Hashem realizes that the Satan convinced Him to judge the Jewish nation. Hashem says, "If the Satan is so powerful that he can convince Me to judge the Jewish nation, then the Jewish people are certainly not guilty of the aveiros they committed. How can they stand up against such a cunning Satan?" With this thought in mind, Hashem abandons the throne of judgment and sits on his throne of compassion.
The Gemara says about the shofar, כיון דמי כבפנים הוא דלזכרון, "Since the shofar arouses Hashem's memory [to remember the good deeds of the Jewish nation], it is as if it was blown inside the Kodesh HaKedashim." The Sfas Emes zt’l quotes this Gemara and explains that when we blow shofar, it is like we are in the Kodesh HaKedashim, and therefore, we must listen to the shofar with immense fear as though we are in the Kodesh HaKedashim.
The Rambam writes, "I, Moshe ben Maimon, when the time for shofar came, took the shofar in my hand, and I thought about Who commanded us to do this mitzvah. My knees knocked against each other from fear, and then I began blowing the shofar."
There are many segulos and brachos that we earn from the mitzvah of tekiyas shofar:
Parnassah
The Shaar HaMelech (3:2) writes, "Before the brachah of tekiyas shofar, cry a lot, especially when you answer amen to the brachah, because that's when it is decided how much money you will earn this year."
The Gemara (Shabbos 117) says, הפת ורדיית שופר מלאכה ואינו היא חכמה, "Blowing shofar and removing bread from the walls of an oven require talent but aren't melachos (on Shabbos)." The Tiferes Shlomo explains that in this statement, the Gemara puts together blowing shofar with taking bread out of the oven to indicate that shofar is mesugal for parnassah.
Children
The Baal HaTanya zy'a taught that the letters after ה"עקר are ר"שופ. This implies that shofar is mesugal for the barren to have children. As the Gemara (Rosh Hashanah 11.) states, וחנה רחל שרה נפקדה השנה בראש, "On Rosh Hashanah, Sarah, Rachel, and Chanah were remembered to bear children."
Zera Kodesh writes, "On Rosh Hashanah, the day the world was created, is a time mesugal for the barren to be remembered to bear children." In the Mussaf Shemonah Esrei we say, לא מי הזה כהיום נפקד. The Imrei Noam zt'l says that נפקד means to be remembered to bear children. He adds that the roshei teivos of פקדנ זהה היוםכ is ך"הנ, and הנך is Hashem's name related to pregnancies, as it states, הרה הנך, "Behold you will be pregnant..." This hints that Rosh Hashanah is when Hashem remembers the barren, blessing them with children.
It states, הארץ בקרב לרוב וידגו, "They shall multiply like fish..." Tzaddikim taught that the gematriya of הארץ בקרב רוב is השנה ראש. This is another indication that Rosh Hashanah is mesugal to bear children.
In תוקף ונתנה we say, יבראון וכמה יעברון כמה. The Tiferes Shlomo zt'l translated these words as follows: יעברון כמה, Hashem judges how many women will enter עיבור, pregnancy, that year and יבראון וכמה, how many people will become בריא, healthy, and cured of their illnesses. The Pnei Menachem zt'l told a yungerman who was waiting several years for children that he should have kavanah by יעברון כמה, for this is a tested and proven segulah for the barren to bear children.
The Apter Rav zt'l (Ohev Yisrael) teaches that when we read שרה את פקד 'וה (that Hashem remembered Sarah to grant her a child), it is mesugal for עקרות פקידת, to bring down the salvation for bearing children.
Refuah
The Mishnah (Rosh Hashanah 3:3) states שופר מאריך. One translation of מאריך is healing, and another translation of מאריך is to live long. Thus, we have a hint that with the shofar,
