When did Yom Kippur first come to be?

When the people broke their word by worshipping a calf of gold, Moses told G‑d:

“Forgive them. And if You do not, erase me from Your book that You have written.”

Meaning:

“I know You want to forgive them. You love them no matter what. You loved them before, when they had all but abandoned You in Egypt. And You love them now as well.”

“What has occurred since then? That you have given them your Torah. So that it’s only this book, your Torah, that stands in the way.”

“And if so, I want no part in such a book.”

So it was that then, through Moses, that G‑d wrote a day of forgiveness into His book.

Because the people are not here for the sake of keeping this Torah. The Torah is here to lift up the people.

It is given to us out of love. Unconditional love. And that is Yom Kippur.

Likkutei Sichot, vol. 21, pp. 173–180; ibid., vol. 9, pp. 237–242.