Mi ne volis respondi tro rapide, do mi unue silentis kaj poste diris, ke ŝi eble pravas.

Because after voli (to want), Esperanto uses the infinitive form of the next verb.

• voli + infinitive
• volis respondi = wanted to answer

So:

• Mi volis respondi = I wanted to answer

Using respondis would wrongly make it a second fully conjugated verb.

Because rapide is an adverb, and it describes the verb respondi.

• rapida = fast, quick (adjective)
• rapide = quickly, fast (adverb)

Here the sentence is talking about how the person would answer, so Esperanto uses the adverb:

• respondi rapide = to answer quickly

Tro means too in the sense of excessively.

So:

• tro rapide = too quickly / too fast

It does not mean just very quickly. If you wanted very, you would use tre:

• tre rapide = very quickly
• tro rapide = too quickly

Because Esperanto normally does not drop subject pronouns the way some languages do.

So in:

• Mi ne volis respondi tro rapide, do mi unue silentis...

the second mi is there because silentis needs an explicit subject.

In English, we can sometimes leave repeated subjects out in looser styles, but standard Esperanto usually keeps them.

Do means so, therefore, or thus.

It shows a result or conclusion:

• Mi ne volis respondi tro rapide, do...
  = I didn’t want to answer too quickly, so...

It is a very common linking word in Esperanto.

Because they tell us when / in what order the actions happened.

• unue = first
• poste = afterward / then / later

They modify the verbs:

• unue silentis = first was silent / first kept quiet
• poste diris = then said

The -e ending is the normal adverb ending in Esperanto.

Yes. Silenti is a normal Esperanto verb meaning to be silent, to keep silent, or to remain quiet.

So:

• silentis = was silent / kept silent

This is natural Esperanto. It does not need an object.

Both commas are normal and helpful.

• Before do: it separates two main clauses.
• Before ke: it marks the start of a subordinate clause.

So the structure is:

• Mi ne volis respondi tro rapide, do mi unue silentis kaj poste diris, ke...

Esperanto punctuation is often fairly straightforward and follows clause boundaries clearly.

Ke means that and introduces a subordinate clause.

So:

• diris, ke ŝi eble pravas
  = said that she may be right

This is very common after verbs like:

• diri = to say
• pensi = to think
• scii = to know
• kredi = to believe

Because ŝi is the subject of pravas.

• ŝi = she
• ŝin = her

In ŝi eble pravas, the woman is the one who is right, so she is the subject, not the object.

Because Esperanto often uses the verb pravi, which means to be right.

So:

• ŝi pravas = she is right

This is very natural Esperanto.
You can also say ŝi estas prava, but ŝi pravas is shorter and more idiomatic in many situations.

Related forms:

• prava = right, correct
• pravi = to be right

Eble means perhaps, maybe, or possibly.

So:

• ŝi eble pravas = she may be right / perhaps she is right

It modifies the whole idea of ŝi pravas.

You could also place it a little differently:

• Eble ŝi pravas

That is also correct. The version in the sentence is very natural.

Because Esperanto does not automatically shift tenses backward the way English sometimes does.

Here:

• diris = said
• pravas = is right

The speaker, at that moment in the past, said that she might be right. The idea of her being right is treated as present relative to that statement, so pravas makes sense.

Esperanto usually keeps the tense that best matches the actual time of the situation, rather than forcing a backshift.

Yes, but the meaning changes.

• Mi ne volis respondi = I did not want to answer
• Mi volis ne respondi = I wanted not to answer / I wanted to refrain from answering

The sentence you were given means the first one: the negation applies to voli.

That is why ne comes before volis.

Because Esperanto allows respondi without stating who or what was answered, if the context is clear.

You can say:

• Mi respondis = I answered
• Mi respondis al ŝi = I answered her / answered to her
• Mi respondis la demandon is generally less standard than using respondi al demando or another phrasing, depending on style

In this sentence, the focus is on not answering too quickly, so no recipient is mentioned.

Esperanto word order is fairly flexible, but this sentence uses a very natural neutral order.

For example:

• ke ŝi eble pravas
• ke eble ŝi pravas

Both are possible.

But the given order is smooth and easy to understand. In Esperanto, word endings do much of the grammatical work, so word order can vary more than in English, though some orders sound more natural than others.