The Amidah (or T'filah) is the central prayer in the three major prayer services of most Jews: morning, afternoon, and evening.
What you are reading is merely a line from it that (in some denominations) changes according to the weather that one would normally experience in the land of Israel (namely from December 4th until the first day of Passover).
It does not mean that Jews believe that God is directly involved in the weather.
In fact, Jews do not believe that God actually listens to prayer. He might, but then again God might not. That is not what prayer is about in Judaism.
It is sort of difficult to talk about prayer in English because we are not using the same word that we use in Hebrew for "prayer." In English, the word "prayer" means to "ask." But the word in Hebrew doesn't mean that.
The Hebrew word t'filah means to judge oneself, to inspect what you are. This is why Jewish prayers consist of fixed words.
Instead of believing that one's prayers are being heard by God or one is asking God for something, like Jehovah's Witnesses do, Jews are doing something entirely different. Jews believe they are mostly hearing God talk to them.
Much of the prayers consist of words from the Psalms and Scripture as well as from the sages that go back to what was said as far back to the days when there was a Temple. When offerings were made, some of these words in the Siddur (where one finds the Amidah) were actually being said, morning, afternoon, and evening. So these words are combined together, blessing God, blessing the time of day, and literally thanking God for where one is in that moment.
There is little in the way of requests in Jewish prayer. Even when one is sick or one dies, prayer is blessing God, asking that one learns to accept their place in the constant changes of reality, blessing these as they come, etc.
Those statements about the weather, for example, are merely used to address the changes one sees in the winter as opposed to the summer when they get to that place in the Amidah.
Why fixed prayers? To learn what we should value, what we should pray for. To be at one with our people, the household of Israel. To ensure that the ideals painfully learned and purified, and for which many have lived and died, shall not perish from the community, and shall have a saving influence upon the individual.--Chaim Stern.