Learn Hardcore Russian: Учитель говорит, что в её стихах простая, но красивая рифма. - The teacher says that in her verses there is a simple but beautiful rhyme.

Учитель говорит, что … = The teacher says that …
It introduces a subordinate clause (reported speech, what the teacher says).
Word-for-word: Учитель говорит, что в её стихах простая, но красивая рифма.
→ The teacher says that there is a simple but beautiful rhyme in her poems.

The question word что? is stressed and usually stands at the beginning of a question:
Что ты сказал? (What did you say?)

In your sentence, что is unstressed and follows говорит, so it is a conjunction, not a question word.

Why is there no word meaning is / there is in в её стихах простая, но красивая рифма?

In Russian, the verb быть (to be) in the present tense is usually omitted in sentences like this.

Understood full form: … что в её стихах (есть) простая, но красивая рифма.
Normal spoken and written Russian: the есть is dropped.

So:

В её стихах простая, но красивая рифма.
literally: In her poems (there is) simple but beautiful rhyme.

This is very common:

В комнате стол. → (There is) a table in the room.
У него машина. → (He) has a car. (literally: At him (there is) a car.)

You only must show быть / есть explicitly when you need to stress existence or contrast:

Там есть рифма, а здесь её нет.
There is rhyme there, but there is none here.

Why is стихах in that form? What case and number is it, and why is it used after в here?

Стихах is:

plural
prepositional case
form of стих (poem, verse)

Full paradigm (singular → plural):

стих (nom.) → стихи
стиха (gen.) → стихов
стиху (dat.) → стихам
стих (acc.) → стихи
стихом (instr.) → стихами
стихе (prep.) → стихах

After в you can get accusative (direction) or prepositional (location):

в её стихи (accusative) = into her poems (direction, movement into) – quite unusual here
в её стихах (prepositional) = in her poems (location, where something is)

We are talking about where the rhyme is, so Russian uses the prepositional: в её стихах.

Why is рифма singular even though стихах is plural (many poems)?

Russian often uses singular for a quality or a general feature that applies to a group of things.

Here, рифма is treated as a general characteristic of her poems:

… в её стихах простая, но красивая рифма.
→ in her poems there is (the) simple but beautiful rhyme; her poetry has simple but beautiful rhyme.

It is similar to English phrases like:

Her poems have beautiful imagery. (uncountable idea)
There is beautiful rhyme in her poems.

If you really wanted to focus on many specific rhymes, you could say:

В её стихах простые, но красивые рифмы.
There are simple but beautiful rhymes in her poems.

But the original sentence sounds more natural: it comments on the overall rhyming style, not on counting individual rhymes.

Why are the adjectives простая and красивая in the feminine singular form? What do they agree with?

In Russian, adjectives agree with the noun they describe in:

gender
number
case

The main noun here is рифма:

рифма is feminine, singular, nominative
so the adjectives must also be: feminine, singular, nominative.

That is why we have:

простая, но красивая рифма
(simple but beautiful rhyme)

If the noun were masculine or plural, the adjectives would change:

простой, но красивый стих (masc.) – a simple but beautiful poem
простые, но красивые стихи (pl.) – simple but beautiful poems

Why is there a comma before что and before но? How does the punctuation work?

There are two main punctuation rules here:

Comma before что
- Учитель говорит, что …
  We have a main clause (Учитель говорит) and a subordinate clause (что в её стихах простая, но красивая рифма).
- Russian normally uses a comma to separate the main clause from the dependent clause introduced by что.
Comma before но
- простая, но красивая рифма
- но (but) links two parts that contrast.
- In Russian, a comma is usually placed before the conjunction но.

So the structure is:

Учитель говорит, (main clause)
что в её стихах (start of subordinate clause)
простая, (first adjective phrase)
но красивая рифма. (second adjective phrase with contrast)

Учитель looks grammatically masculine. How do we know whether the teacher is a man or a woman, and what does её refer to?

The noun учитель is grammatically masculine, but it can refer to a male or female person (especially in everyday speech for school teachers). Gender of the real person is determined by context, pronouns, and agreement.

In the sentence:

Учитель говорит, что в её стихах простая, но красивая рифма.

We see её (her). This её does not grammatically depend on учитель; it just shows that the poems belong to some woman.

Possible readings:

The teacher is talking about some female poet, and её refers to that poet (most natural interpretation).
Less likely from this sentence alone: учитель could be a female teacher talking about her own poems; then we would probably expect в своих стихах (in her own poems) to make that clear.

So:

её = her (possessive pronoun, feminine)
It tells us that the owner of the poems is female.
It does not automatically tell us the teacher’s gender, because учитель itself is masculine by form and neutral for actual gender unless clarified by context.

What is the difference between в её стихах and в своих стихах in this kind of sentence?

Both are prepositional plural, but the pronouns behave differently:

её = her (possessive pronoun for any third-person female)
своих = her own / his own / their own etc. (reflexive possessive, refers back to the subject)

So:

Учитель говорит, что в её стихах простая, но красивая рифма.
- её = some other woman’s poems (a female poet she or he is talking about).
Учитель говорит, что в своих стихах простая, но красивая рифма.
- своих = the subject’s own poems.
- Here, учитель is the one who wrote the poems; it is the teacher’s own poetry.

Russian strongly prefers свой / свои / своих when the owner is the subject of the same clause, to avoid ambiguity.

Could we say простая, но красивая рифма в её стихах instead? How does the word order change the emphasis?

Yes, this word order is grammatically correct, but it has a slightly different emphasis.

Original:

… что в её стихах простая, но красивая рифма.
Emphasis tends to fall on what is in her poems – we start with the location в её стихах, then identify what is there (simple but beautiful rhyme). It is like saying:
- In her poems, there is simple but beautiful rhyme.

Alternative:

… что простая, но красивая рифма в её стихах.
This sounds a bit less natural and may feel incomplete without есть:
- … что простая, но красивая рифма есть в её стихах.

Here the focus moves more onto рифма first (what kind of rhyme), and only then where it is found (in her poems).

In existential sentences (there is / there are), Russian very often starts with a place or context:

В саду красивые цветы. (In the garden there are beautiful flowers.)
На столе книга. (On the table there is a book.)
В её стихах простая, но красивая рифма. (In her poems there is simple but beautiful rhyme.)

Why is the verb говорит (present tense, imperfective) used here, and how would the meaning change with сказал / сказала?

Говорит is:

present tense
imperfective aspect
3rd person singular of говорить (to speak, to say, to tell)

In context, учитель говорит can mean:

What the teacher is saying right now (current speech).
What the teacher generally says as an opinion or repeated statement (habitual).

If we changed it:

Учитель сказал, что в её стихах простая, но красивая рифма.
(masculine)
Учитель сказала, что в её стихах простая, но красивая рифма.
(feminine)

Then:

сказал / сказала is past tense, perfective, a completed act of saying.
It refers to one specific occasion in the past when the teacher said this.

So:

говорит → focuses on ongoing / typical speech.
сказал(а) → focuses on a single completed statement in the past.