Он стесняется говорить с незнакомыми людьми по телефону.

Because the dictionary form is стесняться, a reflexive verb.

• стесняться = to be shy, to feel embarrassed, to feel awkward
• стесняется = he is shy / feels awkward

The ending breaks down like this:

• стесня- = stem
• -ет = 3rd person singular present
• -ся = reflexive marker

So Он стесняется literally means something like He feels shy/awkward.

Not exactly. In this sentence, стесняться is closer to:

• to be shy
• to feel awkward
• to feel self-conscious
• to be embarrassed to do something

So Он стесняется говорить... means he feels uncomfortable or shy about doing it.

It is usually not as strong or moralistic as English to be ashamed.
A better feel here is: He is shy about talking to strangers on the phone.

Because Russian often uses стесняться + infinitive to mean to be shy/embarrassed to do something.

So:

• стесняется говорить = is shy about speaking / is embarrassed to speak

This is a very common pattern in Russian, similar to:

• боится спросить = he is afraid to ask
• любит читать = he likes to read
• хочет уйти = he wants to leave

So the infinitive говорить tells you what action he feels shy about.

Because the preposition с meaning with normally takes the instrumental case.

Here both words are plural, so you get instrumental plural:

• с
  • instrumental
• незнакомыми = instrumental plural of незнакомый
• людьми = instrumental plural of люди

So:

• с незнакомыми людьми = with strangers / with unfamiliar people

That is why it is незнакомыми людьми, not незнакомые люди or незнакомых людей.

It is irregular, and that is normal.

The noun family is:

• человек = person
• люди = people
• людьми = with people / by people

So the plural of человек is not built regularly. You have to learn it as a set:

• nominative plural: люди
• instrumental plural: людьми

This is one of the common irregular noun patterns in Russian.

Because по телефону is the standard Russian expression for by phone / on the phone / over the phone.

A very important grammar point: after по in this expression, Russian uses the dative case:

• телефон = telephone
• по телефону = by phone / on the phone

So телефону here is dative singular, not prepositional.

This phrase is idiomatic. English says on the phone, but Russian says по телефону.

Both can be possible, but говорить is very natural here.

• говорить с кем-то = to talk/speak with someone
• разговаривать с кем-то = to have a conversation with someone

In many contexts they are close, but:

• говорить is broader and often simpler
• разговаривать can sound a bit more specifically like carrying on a conversation

So Он стесняется говорить с незнакомыми людьми по телефону sounds completely normal.

A possible alternative is:

• Он стесняется разговаривать с незнакомыми людьми по телефону

That is also good, just slightly different in feel.

Usually it sounds like a general characteristic or a repeated situation:

• He is shy about talking to strangers on the phone
• He tends to feel awkward talking to strangers on the phone

Because:

• стесняется is present tense
• говорить is imperfective infinitive, which fits a general activity

Depending on context, it could also describe how he feels right now, but without extra context, most learners should understand it as a general trait.

Yes, often it could.

Russian frequently omits subject pronouns when the meaning is clear from context:

• Стесняется говорить с незнакомыми людьми по телефону.

That can work if everyone already knows who is being talked about.

But in an isolated sentence, он is helpful because it makes the subject explicit:

• Он стесняется... = He is shy...

So including он is natural and clear.

Yes. Russian word order is flexible, and changing it usually changes emphasis, not the basic meaning.

The given order is neutral and natural:

• Он стесняется говорить с незнакомыми людьми по телефону.

Other possible orders include:

• Он стесняется по телефону говорить с незнакомыми людьми.
• С незнакомыми людьми он стесняется говорить по телефону.

These alternatives are possible, but they highlight different parts of the sentence.
The original version is a very standard way to say it.

Here it should be one word: незнакомыми.

That is because незнакомый is a normal adjective meaning:

• unfamiliar
• unknown
• not acquainted

So:

• незнакомый человек = a stranger / an unfamiliar person
• с незнакомыми людьми = with strangers

Writing не знакомыми separately would usually be wrong here unless you had a special contrast, which this sentence does not have.