Triumphant, the Jews left Egypt in a swirl of miracles and wonders. Forty days later they stood at the foothills of Mount Sinai to receive the Torah. But while Moses was on Mount Sinai they made a grave mistake: the Sin of the Golden Calf.
As he descended the mountain, Moses saw the people he led out of Egypt dancing around this idol of gold. He dropped the Tablets, shattering them, and after meting out justice, ascended the mountain the next morning to plead with G-d to spare His nation. Forty days later, G-d accepted Moses’ prayers and commanded him to craft a second set of Tablets.
On the first of the month of Elul, the shofar echoed through the Jewish encampment and the nation heard that Moses was ascending to receive the second Tablets. On Mount Sinai Moses devoted his days and nights to learning Torah from G-d, so he could bring it back to the nation. Forty days later, on Yom Kippur, he came down with the second Tablets.
For this reason, these days were set aside as days of Divine forgiveness, a time when G-d is ready to hear and accept our Teshuvah (repentence or return) and is desirous of our closeness. These days culminate in the complete absolution of sin and our reunification with G-d on Yom Kippur.