"And [Moshe] said to them 'I am a hundred and twenty years today, I can no longer go forth or come in..." (Devarim 31:2)
The Gemara, focusing on the word "today", explains this possuk as indicating that Moshe had turned one hundred and twenty on the very day of his death; his birthday and day of departure were the same. This, the Gemara indicates, is the mark of the death of righteous people, as Hashem, "completes their lives from day to day and month to month." It is unclear what exactly is the significance of this. Given a choice of living a long life without "rounding out" the days, or living a shorter life with the days and months "rounded out", would one not rather have the longer life?
The Maharal (Chiddushei Aggados, Sotah 13b) associates another Chazal with this point. He points to a midrash about Sara's life, and uses the verse, "Hashem takes note of the lives of temimim (righteous, literally - complete)"; just as they are "complete" so too are their years "completed". Although it is not clear from the midrash what exactly is referred to by this "completeness", the Maharal compares it to our Gemara, indicating that the point of our Gemara is to delineate the temimus quality of tzaddikim.
But what exactly does this "completeness" mean? It is unreasonable to say that this refers to "complete in the sense of "perfect", for we know the dictum that, "there is no tsaddik in this world who never sins" (Koheles 7:20). So what particular quality of the righteous are we referring to with the metaphor of "complete"?
Indeed, the quality of temimus is not a measure of total righteousness, but rather a perspective on man's mission. One can look at the world simply as a place with many opportunities for mitzvos, which in turn add up to many wonderful rewards. In that perspective man has no particular mission and no all-encompassing accomplishment, but rather mitzvah by mitzvah we attain more and more reward. This frames the world as a sort of carnival where you can earn reward cards through various activities, and more cards you collect the greater your prize.
But there is a much deeper perspective on life. A person's life is a discrete unit of avodas Hashem, and there is a specific mission to be carried out. The subdivisions of time, i.e. years, months, and days, are meaningful subdivisions, each one in turn containing a discrete subdivision of that big picture. Thus, a complete month is a complete unit of avodas Hashem which, together with the other months, forms a complete year. The same is true of days adding up to a month.
A true tzaddik is someone who has this deeper perspective on life. He is not busy collecting mitzvos, but rather fulfilling a clear and distinct mission. An incomplete month, or a partial year, is a total failure from a tzaddik's perspective; one has done good things but has not really "accomplished" anything. Thus, Hashem's reward for the tsaddik is to have his life come to fulfillment - full years, full days.
Moshe Rabbeinu emphasizes this point with the continuation of his statement, "I can no longer go forth or come in." Rashi explains that this can't be referring to physical infirmity, for the passuk emphasizes his total health. Rather, Rashi says, his "wellsprings of wisdom" had dried out.
This does not mean that his mental acuity had dulled. Rather the "wellspring had dried out". There was no longer any new wisdom coming forth from his studies. For the true tzaddik who finds meaning in the unique mission that each day and month brings, a simple repetition of "what was, will be" is contrary to his approach to life.
As we begin our new year iy"H, we must bear in mind that a new unit of time is not just a yardstick for "how much more of the same will we have". Rather it is a new phase of avodas Hashem, with a starting and ending point at Rosh Hashana. Our job is to undertake the new year as a new mission and iy"H Hashem will give us the opportunity to wholly complete the mission of our lives, so that when we finish our mission in this world, it will be with a sense of completeness: days, months, and years.