Mi hermano empieza a tartamudear cuando habla en público.

Because mi without an accent is the possessive adjective meaning my.

• mi hermano = my brother
• mí with an accent is a pronoun meaning me, usually used after a preposition:
  • para mí = for me
  • a mí = to me

So in this sentence, mi is correct because it shows possession.

Because empezar is a stem-changing verb in the present tense.

In the present indicative, the e in the stem changes to ie in most forms:

• yo empiezo
• tú empiezas
• él/ella empieza
• nosotros empezamos
• vosotros empezáis
• ellos empiezan

So mi hermano empieza means my brother starts / begins.

This kind of change is very common in Spanish. Other verbs like this include pensar → piensa and cerrar → cierra.

Because empezar is commonly followed by a + infinitive when it means to begin/start doing something.

So:

• empezar a + infinitive
• empieza a tartamudear = he starts to stutter

This is a fixed pattern in Spanish. Some similar examples:

• Empiezo a trabajar a las ocho. = I start working at eight.
• Empezaron a reírse. = They started laughing.

You normally would not say empieza tartamudear.

There is a small difference in meaning.

• tartamudea = he stutters / he is a person who stutters
• empieza a tartamudear = he starts to stutter

The sentence is focusing on the moment or reaction: when he speaks in public, he begins to stutter.

So this version suggests that public speaking triggers the stuttering, rather than simply describing a permanent characteristic.

In this sentence, cuando is being used to talk about something that happens habitually or generally, so Spanish uses the present indicative:

• cuando habla en público = when he speaks in public

This means whenever he speaks in public.

You would use the subjunctive after cuando if you were talking about a future or not-yet-realised event:

• Cuando hable en público, estará nervioso.
  = When he speaks in public, he’ll be nervous.

So:

• cuando habla = repeated/general situation
• cuando hable = future situation

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

• habla already tells us it is he/she/you formal
• In this sentence, the subject is understood from the context: mi hermano

So cuando habla en público naturally means when he speaks in public.

You could say cuando él habla en público, but it usually sounds unnecessary unless you want emphasis or contrast:

• Mi hermano empieza a tartamudear cuando él habla en público, pero yo no.

It can correspond to both in English, depending on context, but the most natural translation here is:

• when he speaks in public

Spanish often uses the simple present where English might use either:

• when he speaks in public
• when he is speaking in public

In this sentence, it describes a general repeated situation, so when he speaks in public is the best fit.

En público means in public or before an audience, depending on context.

Here it means that he is speaking in front of other people, not privately.

A useful point: Spanish often uses en público without an article.

• hablar en público = to speak in public
• aparecer en público = to appear in public

So you should learn en público as a set expression.

Yes. Tartamudear is the standard verb meaning to stutter or to stammer.

Related words:

• tartamudear = to stutter
• tartamudeo = stutter / stammering
• tartamudo / tartamuda = a person who stutters

Example:

• De niño, tartamudeaba mucho. = As a child, he stuttered a lot.

In Spain, tartamudear is completely normal and widely understood.

A few useful pronunciation points:

• hermano: the h is silent, so it sounds roughly like er-MA-no
• empieza: roughly em-PYE-tha in much of Spain, or em-PYE-sa in Latin America
• tartamudear: roughly tar-ta-mu-de-AR

Stress:

• herMAno
• emPIEza
• tartamuDEAR

For Spain specifically, many speakers pronounce z like the th in thin, so empieza often sounds like em-PYE-tha.

Yes, that is grammatical, but it means something slightly different.

• Mi hermano empieza a tartamudear cuando habla en público
  = My brother starts to stutter when he speaks in public
• Mi hermano tartamudea cuando habla en público
  = My brother stutters when he speaks in public

The first version highlights the onset of the stuttering.
The second simply states that he does stutter in that situation.

Both are natural; the choice depends on the nuance you want.

Yes, Spanish word order is fairly flexible, though the original is the most neutral and natural.

For example:

• Cuando habla en público, mi hermano empieza a tartamudear.

This means the same thing, but it puts the time/situation first: When he speaks in public, my brother starts to stutter.

The original sentence is a very standard, neutral way to say it.