Al principio de la entrevista, tuve que respirar hondo para mantener la calma.

Because al is the contraction of a + el.

• a el principio → al principio
• This is required in normal Spanish whenever a and el come together.

So al principio de la entrevista literally comes from a el principio de la entrevista.

A similar example:

• Voy al supermercado = I’m going to the supermarket

They are very similar, and in this sentence both would sound natural.

• al principio = at the beginning / at first
• al comienzo = at the beginning / at the start

In many contexts they are interchangeable:

• Al principio de la entrevista...
• Al comienzo de la entrevista...

That said, al principio is extremely common and often the most natural choice in everyday speech.

Because principio needs a complement here: the beginning of what?

• el principio de la entrevista = the beginning of the interview

Spanish often uses de where English uses of:

• el final de la película = the end of the film
• el inicio del curso = the start of the course

Here, la entrevista is specific, so the article la is needed.

This is a very common question because both can mean had to, but they are used differently.

tuve que

This is the preterite of tener que and usually refers to a completed, specific situation:

• Tuve que respirar hondo = I had to take a deep breath

It sounds like a concrete action at that moment in the interview.

tenía que

This is the imperfect and usually suggests:

• an ongoing situation
• background information
• repeated obligation
• something seen as less bounded

For example:

• Durante esa época, tenía que hacer muchas entrevistas.
  = During that time, I had to do a lot of interviews.

So in your sentence, tuve que is used because it refers to one specific moment and one specific need.

Because Spanish often drops subject pronouns when the verb already makes the subject clear.

• tuve already tells us the subject is yo
• So (yo) tuve que respirar hondo is normal

You could include yo for emphasis or contrast:

• Yo tuve que respirar hondo, pero ella no.

But without emphasis, leaving it out is more natural.

It expresses obligation or necessity: to have to do something.

Structure:

• tener que + infinitive

Examples:

• Tengo que estudiar = I have to study
• Tuve que salir = I had to leave
• Tendré que esperar = I’ll have to wait

So:

• tuve que respirar hondo = I had to take a deep breath

Respirar hondo is the usual idiomatic expression for to take a deep breath / breathe deeply.

Even though hondo is basically an adjective meaning deep, in this expression it works in a very natural fixed phrase:

• respirar hondo
• inspirar hondo

You can say respirar profundamente, and it is grammatically correct, but it sounds less idiomatic for this specific idea.

So if you want the natural everyday expression, use:

• respirar hondo

Because para + infinitive is a very common way to express purpose in Spanish.

It answers the question what for?

• Tuve que respirar hondo para mantener la calma.
• I had to take a deep breath in order to stay calm.

More examples:

• Estudio para aprender. = I study in order to learn.
• Fui temprano para evitar problemas. = I went early to avoid problems.

So here, para mantener la calma explains the purpose of breathing deeply.

Because in Spanish, abstract nouns often take the definite article where English would not.

So:

• mantener la calma = keep calm / remain calm

This is the standard expression. Without la, it would sound wrong here.

Other similar examples:

• tener paciencia is common without an article, but
• perder la paciencia uses the article
• guardar la calma
• recuperar la confianza

Spanish article use does not always match English, so this is something you often just learn as part of the expression.

Yes, but the nuance is slightly different.

• para mantener la calma = to keep calm / to remain calm
• para calmarme = to calm myself down

In your sentence, mantener la calma suggests the speaker was trying not to lose composure.
calmarme suggests the speaker was already becoming nervous and wanted to reduce that feeling.

Both are natural, but they are not exactly identical.

Because Al principio de la entrevista is an introductory time phrase, and a comma helps separate it from the main clause.

• Al principio de la entrevista, tuve que respirar hondo...

In Spanish, this comma is very natural and often recommended. Sometimes short introductory phrases appear without a comma, but here the pause feels clear and standard.

Yes. Entrevista is a broad word and can refer to many kinds of interviews, such as:

• a job interview
• a media interview
• an interview for research
• a formal meeting where someone is questioned

Usually the context tells you which kind it is. If needed, Spanish can be more specific:

• entrevista de trabajo = job interview
• entrevista periodística = press/media interview

So la entrevista by itself is perfectly normal if the context is already clear.