Our Sages found a hint in the verse "I am for my Beloved, and my Beloved is for me" (Shir Hashirim 6:3), where the first letters of each word form the Hebrew word "Elul" The final letters of each word, in gematria, equal forty, symbolizing that during these forty days between Elul , Rosh Haschana and Yom Kippour, a person should draw close to their Beloved (GOD), and in turn, their Beloved will draw close to them and accept their repentance with love.
These days are days of divine favor, going back to ancient times. When Israel sinned with the Golden Calf and the Tablets were broken, Moses ascended the mountain and pleaded before GOD for mercy and forgiveness for the sin of the Golden Calf. GOD was appeased and said, "Carve for yourself two stone tablets like the first ones." Moses ascended the mountain on Rosh Chodesh Elul and stayed there for forty days, until Yom Kippur. On Yom Kippur, he descended the mountain with the Tablets, which GOD gave to the people of Israel with goodwill and joy. These forty days were established for generations as days of divine favor, repentance, and atonement.
From the first of Elul until the day before Rosh Hashanah, the shofar is sounded daily with four sounds: Tekiah, Shevarim, Teruah, Tekiah.
This shofar blowing is not a halachic obligation, but rather a Jewish custom. When Moses ascended Mount Sinai on Rosh Chodesh Elul to receive the second Tablets, the shofar was sounded in the camp to inform all the people of Israel that Moses had ascended to the heavens, so they would not err again by following idolatry. Therefore, the custom was established to blow the shofar starting from Rosh Chodesh Elul to remind us of Moses' ascent and to signify that the people of Israel repented for the sin of the Golden Calf and were granted forgiveness, and the second Tablets were given to them with goodwill at the end of the forty days.
In the month of Elul, the last month of the Hebrew calendar, a significant process occurs for every personโa process of awakening and repentance in preparation for the upcoming holidays, particularly Rosh Hashanah and Yom Kippur. There are five ways for a person to renew their inner connection with GOD: Torah, prayer, charity, repentance, and bringing the redemption closer. All of this is done with love for GOD, love for Israel, and love for the World.
(From Rabbi Yehuda Roven)