A professora explicou a gramática antes de corrigir a redação.

Because professora is a feminine noun, meaning female teacher. In Portuguese, articles usually match the gender of the noun:

• o professor = the male teacher
• a professora = the female teacher

The ending -a often signals a feminine form, though not always.

Explicou is the preterite form of explicar. It means explained in the sense of a completed action in the past.

Here it is:

• eu expliquei = I explained
• você/ele/ela explicou = you/he/she explained

So A professora explicou... means the teacher completed the explanation.

Explicou is the preterite, which usually presents the action as finished and complete.

• explicou = explained, completed that action
• explicava = was explaining / used to explain / explained repeatedly or as background

In this sentence, the teacher first explained the grammar and then corrected the essay, so the completed-action form explicou fits well.

In Portuguese, it is very common to use the definite article with nouns where English may or may not use the.

So:

• a gramática = the grammar

In this sentence, it likely refers to the grammar being discussed in that situation, such as a lesson point or the grammar related to the writing assignment.

Portuguese often sounds more natural with the article here than English would.

For the same reason: redação is a noun, and Portuguese often uses the definite article with direct objects when they are understood as specific.

• corrigir a redação = correct the essay / the composition

This suggests a particular piece of writing, not just essay-writing in general.

Redação often means a school composition, essay, or writing assignment. In Brazilian Portuguese, it is very commonly used in educational contexts.

Depending on context, it can also relate more broadly to writing or wording, but in a sentence like this, essay/composition is the most natural meaning.

Because Portuguese uses the preposition de after antes when it is followed by a verb.

• antes de corrigir = before correcting
• antes da aula = before class

So antes de + infinitive is a very common structure:

• antes de sair = before leaving
• antes de estudar = before studying

After antes de, Portuguese normally uses the infinitive when expressing an action like before doing something.

So:

• antes de corrigir a redação = before correcting the essay

This works much like English before correcting.

If you want to make the subject very explicit, Portuguese can also use a different structure, such as:

• antes de ela corrigir a redação = before she corrected / before she would correct the essay

But in your sentence, the simpler infinitive structure is the normal choice.

Because corrigir is a verb, not a noun with an article.

Portuguese contracts de + o/a:

• de + o = do
• de + a = da

But here we have:

• de corrigir

Since corrigir is just the infinitive verb to correct, there is nothing to contract.

Yes. Portuguese allows some flexibility. You could also say:

• Antes de corrigir a redação, a professora explicou a gramática.

This means the same thing. The difference is mainly in focus and rhythm:

• A professora explicou... antes de corrigir... = starts with the teacher’s action
• Antes de corrigir..., a professora explicou... = starts with the time sequence

Both are natural.

They show stress and, sometimes, vowel quality.

• gramática: the stress falls on má
  Pronounced roughly gra-MA-ti-ca
• redação: the -ção ending is stressed
  Pronounced roughly re-da-SÃO

The ã in -ção is nasal, which is very characteristic of Portuguese pronunciation.

Not exactly.

• explicar = to explain
• ensinar = to teach

In many contexts they overlap, but they are not identical.

• A professora explicou a gramática. = The teacher explained the grammar.
• A professora ensinou gramática. = The teacher taught grammar.

Explicar focuses on making something clear.
Ensinar focuses more on the broader act of teaching.

It could suggest either one depending on context, but in this sentence it most naturally means the grammar content/rules that the teacher explained before correcting the essay.

If someone wanted to talk more generally about the school subject, they might still use gramática, but here the sentence sounds like a concrete classroom action: first she explained the grammar, then she corrected the writing.