Learn Hardcore Russian: Сезон дождей становится прохладным.

It literally translates to "season of rains." The noun дождей is the genitive plural of дождь and is used here to specify what type of season it is—that is, a season characterized by rain. In Russian, modifying nouns that describe a characteristic or a possession take the genitive case.

Why is the adjective прохладным in the instrumental case rather than the nominative?

When a Russian sentence uses a linking verb like становиться (to become), the predicate adjective describing the new state is placed in the instrumental case. Thus, прохладным is in the instrumental to indicate the state that Сезон дождей is becoming.

What is the function of the verb становляется in this sentence, and what tense and aspect does it express?

Становляется is the third-person singular form of становиться, meaning "to become." It is in the present tense and is an imperfective verb, which conveys an ongoing or gradual change of state, linking the subject to its new attribute.

How do the cases in Сезон дождей and прохладным work together grammatically?