Răspundem fie la telefon, fie prin email.

Răspundem is the 1st person plural present tense of a răspunde.

So:

• eu răspund = I answer / I respond
• noi răspundem = we answer / we respond

In this sentence, răspundem means we answer or we respond.

Romanian often drops subject pronouns when they are not needed, because the verb ending already shows who the subject is.

So:

• răspundem already tells you it means we answer/respond
• adding noi would be possible, but only if you want emphasis

Compare:

• Răspundem fie la telefon, fie prin email. = We answer either by phone or by email.
• Noi răspundem... = We answer... (more emphasis)

Fie ... fie ... means either ... or ....

So:

• fie la telefon, fie prin email = either by phone or by email

This structure is common in Romanian and is often a bit more formal or careful than just using sau.

For example:

• Venim fie luni, fie marți. = We are coming either Monday or Tuesday.
• Poți plăti fie cash, fie cu cardul. = You can pay either in cash or by card.

Yes, you could say:

• Răspundem la telefon sau prin email.

That is also correct and natural.
But fie ... fie ... highlights the two alternatives more clearly, like English either ... or ....

So the difference is roughly:

• sau = or
• fie ... fie ... = either ... or ...

This is mainly about Romanian usage with different prepositions.

• la telefon is the normal Romanian expression for on the phone / by phone
• prin email means through email / by email

These are idiomatic combinations, so they are best learned as chunks:

• a răspunde la telefon = to answer on the phone / to answer the phone
• a răspunde prin email = to reply by email

Romanian does not always match English prepositions exactly.

Here la telefon means by phone or on the phone.

It is a very common expression. Depending on context, a răspunde la telefon can mean:

• to answer the phone
• to respond by phone
• to deal with something over the phone

In your sentence, because it is paired with prin email, the meaning is clearly by phone.

Because Romanian often uses nouns without the article in general expressions after prepositions.

So:

• la telefon
• prin email

sound natural as general means of communication.

This is similar to how English says:

• by phone
• by email

rather than by the phone or by the email.

Not always. This is an important point.

A răspunde can work in more than one pattern:

1. a răspunde la ceva = to answer/respond to something

   • răspund la întrebare = I answer the question
   • răspund la telefon = I answer the phone / respond by phone

2. a răspunde cuiva = to answer someone

   • îi răspund Mariei = I answer Maria / I reply to Maria

So:

• la is used with the thing
• the person is usually in the dative

Yes, email is widely used in Romanian. You may also see:

• e-mail
• sometimes mail in informal speech

In modern everyday Romanian, email is very common and easy to understand.
So prin email is perfectly normal.

You might also hear alternatives such as:

• prin e-mail
• pe email in some informal usage

But prin email is clear and standard enough for learners.

A helpful approximation is:

ruh-SPOON-dem fee-eh la tele-FON, fee-eh preen ee-MAIL

A few details:

• ă in Răspundem is a short central vowel, similar to the a in English about
• u in spun is like oo in good, not like you
• fie is usually pronounced in two syllables: fi-e
• prin sounds like preen
• telefon has stress on the last syllable: tele-FON

Yes. You can switch the two alternatives:

• Răspundem fie prin email, fie la telefon.

That still means the same thing.

Romanian word order is fairly flexible, but the version you were given is very natural. The main thing is to keep the paired structure:

• fie X, fie Y

It sounds neutral to slightly formal.

Why?

• răspundem is standard and neutral
• fie ... fie ... is a little more structured and formal than simple sau
• prin email is modern and normal

So this would fit well in:

• customer service language
• business communication
• website information
• polite general statements

A more casual version might be:

• Răspundem la telefon sau pe email.

But your original sentence is very good standard Romanian.