On the last day of Moshe’s life, he said, “Today I am 120 years old.” The Talmud interprets the word “today” to mean that it was his birthday. “This teaches,” says the Talmud, “that G‑d sits and completes the years of the righteous from day to day and month to month.”
What is the significance of the righteous passing away on their birthdate?
The Tanya teaches that the soul of a tzaddik, a righteous person, can influence his disciples even more so after his passing than when he was alive. For in the tzaddik’s lifetime, when his soul was contained within a physical body, only a glimmer of his soul was able to radiate beyond his body, reaching his disciples only through his thoughts and words. After his passing, however, the tzaddik is no longer limited in this way. Moreover, the ascent of the tzaddik’s soul to its source causes sublime spiritual energy to radiate upon all whom he influenced to become servants of G‑d, instilling thoughts of repentance and good deeds in their hearts.
Thus, in a sense, the day of the tzaddik’s passing completes his birth. For on the day of his passing, his gift to the world is fully revealed and effective, whereas on the day he was born it was only in potential. G‑d therefore “completes the years of the righteous from day to day and month to month,” because a tzaddik’s yahrtzeit is synonymous with his birthday.
—Likkutei Sichos, vol. 16, p. 350