По её голосу я понял, что она устала.

Because по here normally takes the dative case, and the dative singular of голос is голосу.

• nominative: голос = voice
• dative: голосу

So:

• по её голосу = by her voice / from her voice / judging by her voice

This is a very common Russian pattern:

• по глазам = by someone’s eyes
• по виду = by someone’s appearance
• по тону = by someone’s tone

Here по means something like:

• by
• from
• judging by
• based on

So По её голосу я понял... means:

• From her voice, I understood...
• Judging by her voice, I realized...

This is not the basic spatial meaning of по like along or on. It is a different common use of the preposition.

In this sentence, её is a possessive pronoun meaning her.

Russian её in this meaning does not change by case:

• её голос = her voice
• по её голосу = by her voice
• с её голосом = with her voice

So even though голосу changes to dative, её stays её.

A learner should also know that её can mean her or hers, depending on context, but here it clearly means her.

It is pronounced roughly ye-yo.

The two dots on ё matter: they show that this is a yo sound.

So:

• её ≈ yi-YO or ye-YO, depending on how detailed you want to be

In normal printed Russian, ё is sometimes written as е, but here it is helpful that it is written correctly as ё.

Because Russian past tense verbs agree with the subject in gender and number.

• я понял = I understood / I realized (male speaker)
• я поняла = I understood / I realized (female speaker)

So this sentence assumes the speaker is male. If the speaker were female, it would be:

По её голосу я поняла, что она устала.

Because in the past tense, Russian verbs also agree with the subject’s gender.

• он устал = he got tired / was tired
• она устала = she got tired / was tired
• оно устало = it got tired / was tired
• они устали = they got tired / were tired

Since она means she, the verb must be устала.

Because понял is perfective, and here it means a completed realization:

• я понял = I realized / I understood at that moment

This fits the meaning well: hearing her voice led to a specific conclusion.

By contrast, понимал is imperfective and usually suggests an ongoing state or repeated understanding:

• я понимал = I understood / I was understanding

In this sentence, Russian prefers понял because it focuses on the moment of realization.

Because что она устала is a subordinate clause.

Russian normally puts a comma before что when it introduces a clause:

• Я знаю, что он дома.
• Она сказала, что придёт.
• Я понял, что она устала.

So the comma is required here.

Что here means that and introduces what was understood:

• я понял = I understood / I realized
• что она устала = that she was tired

So the structure is:

• I realized that...

This is one of the most common uses of что in Russian.

Yes, both are possible.

• По её голосу я понял, что она устала.
• Я понял по её голосу, что она устала.

Both mean essentially the same thing.

The version with По её голосу first puts more emphasis on the clue or evidence: from her voice.

Russian word order is more flexible than English, so moving parts of the sentence often changes emphasis rather than basic meaning.

In this sentence, was tired is usually the best natural translation.

The verb устать is perfective and literally has the sense to become tired / to get tired, but in many contexts the English result state sounds more natural:

• Я понял, что она устала.
  = I realized that she was tired.

So grammatically the Russian verb comes from устать, but in English you will often translate it with was tired, because that is the state the speaker notices.

Yes, especially in conversation or informal style:

• По её голосу понял, что она устала.

Russian can omit a subject pronoun when it is clear from context. But the full version with я is completely normal and often better for learners because it is clearer.

Not in this meaning.

If you want to say judging by her voice or from her voice, the natural Russian expression is по её голосу.

• из её голоса is not idiomatic here
• от её голоса would mean something very different and does not work for this idea

So this is a set Russian pattern:

• по голосу
• по тону
• по выражению лица
• по глазам

A fairly literal breakdown is:

• По её голосу = By/from her voice
• я понял = I understood / I realized
• что = that
• она устала = she was tired

So very literally:

By her voice I understood that she was tired.

More natural English would usually be:

From her voice, I realized that she was tired.