Learn Hardcore Spanish (Latin America): Cuando leo su poema en voz alta, admiro lo musical que es su poesía. - When I read her poem out loud, I admire how musical her poetry is.

leo already shows the subject is yo (1st person singular).
Saying yo leo is possible, but it sounds more emphatic: I (as opposed to someone else) read.

So Cuando leo su poema… naturally means When I read his/her poem…

Why is it cuando leo and not cuando lea?

Cuando can be followed by either indicative or subjunctive, depending on meaning.

Indicative (leo): used for habitual actions, general truths, or things considered real.
- Cuando leo su poema, admiro… = Whenever I read his poem (any time I do this), I admire…
Subjunctive (lea): used for future, uncertain, or hypothetical events.
- Cuando lea su poema, te digo qué pienso. = When I read his poem (in the future), I’ll tell you what I think.

In your sentence, it’s a repeated/typical situation, so leo (indicative) is correct.

What does en voz alta mean exactly? Can I just say alto?

En voz alta literally means in a loud voice, and idiomatically it means out loud / aloud.

leer en voz alta = to read out loud
It contrasts with leer en silencio = to read silently.

You generally don’t just say leer alto in standard Spanish for “read out loud.” The natural, common expression is leer en voz alta.

What’s the difference between poema and poesía here?

poema = a specific poem (a single text or composition).
poesía = poetry in general, or someone’s poetic style/overall work.

In the sentence:

su poema = that particular poem of his/hers.
su poesía = their poetry as an art form or style.

So the idea is: When I read this specific poem out loud, I admire how musical their poetry (overall) is.

Why is su used twice, and what exactly can su mean? Could I use tu instead?

Su is a possessive adjective that can mean:

his
her
your (formal usted)
their
your (plural ustedes, in Latin America)

Which one it means depends on context, not on form. Here, su poema and su poesía refer to the same owner (who could be he, she, usted, they, etc.).

You can use tu only if you’re speaking informally to tú:

Cuando leo tu poema en voz alta, admiro lo musical que es tu poesía.

In much of Latin America, usted is common even in somewhat informal situations, so su is frequent.

What does lo mean in lo musical que es su poesía?

Here lo is the neutral article, not a pronoun. It turns the adjective musical into an abstract noun phrase:

lo musical ≈ the musical quality / how musical (it is)

The structure lo + adjective + que + verb is very common:

Me sorprende lo difícil que es. = I’m surprised how difficult it is.
No sabes lo caro que está todo. = You have no idea how expensive everything is.

So admiro lo musical que es su poesía = I admire how musical his/her poetry is.

Is musical an adjective here? Why doesn’t it agree with poesía in gender and number?

Yes, musical is an adjective, but in lo musical it’s part of a neutral expression:

lo + adjective → “the … thing / the … quality / what is …”

Because lo is neutral and not masculine or feminine, the adjective stays in its base form, not matching any noun:

lo bueno (the good part/thing)
lo interesante (what is interesting)
lo musical (the musical aspect/quality)

So it’s not directly modifying poesía; it’s part of an abstract phrase: the musical-ness of his/her poetry.

Could I say admiro cuán musical es su poesía or admiro qué tan musical es su poesía? Are they the same as lo musical que es su poesía?

They’re very close in meaning, but there are nuances:

Admiro lo musical que es su poesía.
- Very natural and common.
- Slightly more neutral and idiomatic.
Admiro cuán musical es su poesía.
- Correct, sounds a bit more formal or literary in much of Latin America.
Admiro qué tan musical es su poesía.
- Also used, especially in Latin America, and feels quite natural in speech.

All three can be understood as I admire how musical his/her poetry is, but lo + adjective + que is extremely common in everyday Spanish.

Why is it admiro and not me admiro?

In Spanish, admirar is a regular transitive verb when it means “to admire” someone/something:

Admiro su poesía. = I admire his/her poetry.
Admiro lo musical que es su poesía. = I admire how musical his/her poetry is.

Admirarse exists as a reflexive verb, but it usually means to be amazed/surprised (often at something unexpected), not “to admire aesthetically”:

Me admiro de lo que pasó. = I’m amazed at what happened.

Here you simply want admiro, not the reflexive form.

Could I say admiro lo musical que su poesía es, with the noun before es?

No, that word order sounds wrong in Spanish. In this structure:

lo + adjective + que + [subject] + [verb]

the normal order is:

lo musical que es su poesía
(literally: how musical is his/her poetry)

Switching the subject and verb here (lo musical que su poesía es) is ungrammatical in standard Spanish. Keep:

lo + adjective + que + verb + subject → lo musical que es su poesía

Could I say Al leer su poema en voz alta, admiro lo musical que es su poesía instead of Cuando leo…? Is there any difference?

Yes, you can, and both are correct, but there’s a nuance:

Cuando leo su poema…
- Emphasizes the time/occasion: whenever/when I read his poem… (habitual).
Al leer su poema…
- Literally On/while reading his poem…
- Slightly more formal or written style; focuses a bit more on the action itself (upon reading).

In everyday speech, cuando leo is more common and natural; al leer sounds a bit more formal or bookish.