La profesora me explicó la manera de usar el diccionario.

Because profesora is the feminine form of profesor. It tells you the teacher is female.

• el profesor = the male teacher
• la profesora = the female teacher

In modern Spanish, job titles often change form to match the person's gender.

Me means to me.

So La profesora me explicó... literally means The teacher explained to me...

It is an indirect object pronoun, and in Spanish these pronouns usually go before the conjugated verb:

• me explicó = explained to me
• te explicó = explained to you
• le explicó = explained to him/her

Because in Spanish, unstressed object pronouns normally come before a conjugated verb.

So Spanish says:

• me explicó

not:

• explicó me

This is a very common pattern:

• Me dijo la verdad. = He/She told me the truth.
• Te mandé un mensaje. = I sent you a message.

Explicó is the preterite (simple past), third person singular, from explicar.

It means:

• he explained
• she explained
• it explained
  depending on context

Here it means she explained, because the subject is la profesora.

So:

• explica = explains / is explaining
• explicó = explained

The accent mark shows where the stress goes:

• explicó → stress on the last syllable: ex-pli-CÓ

It also helps distinguish it from explico:

• explico = I explain
• explicó = he/she explained

So the accent is important both for pronunciation and meaning.

La manera de usar means the way of using or more naturally in English, the way to use / how to use.

This structure is very common in Spanish:

• la manera de + infinitive
• la forma de + infinitive

Examples:

• la manera de hacerlo = the way to do it
• la forma de llegar = the way to get there

In this sentence, la manera de usar el diccionario means how to use the dictionary.

Because manera normally takes de before an infinitive.

So Spanish says:

• la manera de usar

not:

• la manera usar

Think of it as part of the pattern:

• la manera de estudiar
• la forma de cocinar
• una forma de aprender

After many nouns, Spanish uses de + infinitive where English might use to + verb.

Yes. A very natural alternative is:

• La profesora me explicó cómo usar el diccionario.

This is often even more natural in everyday Spanish.

Both are correct:

• me explicó la manera de usar el diccionario
• me explicó cómo usar el diccionario

The version with cómo sounds a little more direct and conversational.

Spanish uses articles very often, more often than English does.

Here el diccionario means the dictionary. It may refer to:

• a specific dictionary already understood from context, or
• the dictionary as the tool being discussed

Spanish usually prefers the article in cases like this, so usar el diccionario sounds normal.

Yes. In this sentence:

• La profesora = subject
• me = indirect object (to me)
• explicó = verb
• la manera de usar el diccionario = direct object

So the structure is:

[Subject] + [Indirect object pronoun] + [Verb] + [Direct object]

Literally:

The teacher + to me + explained + the way of using the dictionary

Sometimes yes, but the meaning changes slightly.

• me explicó = she explained to me
• me enseñó = she taught me / showed me

Explicar focuses on giving an explanation.
Enseñar focuses more on teaching or showing.

So:

• La profesora me explicó cómo usar el diccionario. = The teacher explained how to use the dictionary.
• La profesora me enseñó a usar el diccionario. = The teacher taught me how to use the dictionary.

Both are possible, but they are not exactly identical.

There is no separate a before me because the indirect object is already expressed by the pronoun me.

Also, la manera is a thing, not a person, so there is no personal a.

Compare:

• La profesora me explicó la lección. = no personal a
• La profesora vio a María. = personal a because María is a person

So in your sentence, no extra a is needed.

A very natural everyday version would be:

• La profesora me explicó cómo usar el diccionario.

Another possibility:

• La profesora me enseñó a usar el diccionario.

Your original sentence is correct, but cómo usar is often the most natural choice in conversation.