The Mishna explains that in Rosh haShana God Almighty examines, based on our deeds and merits, if He would grant us another year of life. We have a window of opportunity from Rosh haShana to Yom Kippur to admit our mistakes and appeal an eventual negative decree, confessing, asking for forgiveness, etc.At the time of the Neila, the decree is sealed and hopefully we were sealed for life.
Assuming God granted us the gift of life and good health to make it to the next year, a question still remains: will we be granted additionally 'rain', in other words, the material means to maintain a life of dignity?
In Sukkot, our rabbis explain, God Almighty determines 'the sentence for the water /rain' (nidonim al hamaim). In Jewish tradition 'rain' represents the source of livelihood (parnasa). Man works the land, sows the seeds and watches his field, but ultimately, it is the God given rain what will determine the success of the harvest. 'Rain' is the most critical element for nutrition and livelihood.
This is why at the beginning of the new season (autumn/winter), from the Musaf prayer of Shemini Atzeret, we change the words morid hatal (You make the dew descend) for the prayer of the rain mashib haruach uMorid haGeshem, "You make the wind blow, and the rain descend."
It is important to notice that at this specific point we are still not asking for rain (tomorrow, BH, we will see why). We begin by praising God for His power and kindness. When saying mashib haruach we are acknowledging God as the source of rain, and by extension, the ultimate source of our livelihood.