Learn French: Soudain, il s'est mis à pleuvoir.

The base verb mettre means to put, but the pronominal form se mettre à is an idiomatic expression meaning to start or to begin doing something. You can imagine it as putting oneself to a task, though in this sentence it is used impersonally for the weather.

Since il just means an impersonal "it" for the weather, why do we still use the reflexive pronoun s'?

Even when a subject is the impersonal weather il, the verb must keep its required grammar. Because the idiomatic verb is strictly se mettre à, the reflexive pronoun se (which contracts to s' before a vowel) must remain, even if no actual person is doing the action.

Why does this use est instead of a for the passé composé? Doesn't mettre usually take avoir?

You are right that the standard verb mettre takes avoir (as in il a mis). However, all pronominal verbs in French require the auxiliary verb être in compound tenses. Since the verb here is se mettre, it must be conjugated as il s'est mis.

Why is there an à before pleuvoir?