Учительница сказала, что скрипач должен лучше чувствовать ритм и держать такт.

Учительница means female teacher, while учитель means male teacher or sometimes just teacher in a more general dictionary sense.

In this sentence, the speaker specifically refers to a woman, so учительница is used. That also affects the verb later: сказала is feminine because the subject is feminine.

In Russian, the past tense agrees with the subject in gender and number.

• сказал = he said
• сказала = she said
• сказало = it said
• сказали = they said

Since учительница is feminine, the verb is сказала.

Что here means that and introduces a subordinate clause.

So:

• Учительница сказала = The teacher said
• что скрипач должен... = that the violinist should / must...

Russian often uses что the same way English uses that after verbs like say, know, think, and so on.

Yes. Скрипач means violinist or fiddler.

It comes from скрипка = violin.
A native English speaker may notice that Russian often has specific nouns for musicians based on their instrument:

• пианист = pianist
• гитарист = guitarist
• скрипач = violinist

In this sentence, скрипач is the subject of the subordinate clause: the violinist should...

Должен means something like must, should, or is supposed to, depending on context.

Here it expresses obligation or necessity:

• скрипач должен лучше чувствовать ритм = the violinist should/must feel the rhythm better

Like past tense, должен changes for gender and number:

• должен = masculine singular
• должна = feminine singular
• должно = neuter singular
• должны = plural

Because скрипач is masculine singular, we use должен.

After должен, Russian normally uses the infinitive.

So the pattern is:

• кто-то должен + infinitive
• someone must/should do something

Here:

• должен чувствовать = should feel
• должен держать = should keep

This is very common in Russian:

• Я должен работать = I must work
• Она должна учиться = She must study

Лучше is the comparative form of хорошо and means better.

So:

• хорошо = well
• лучше = better

In Russian, лучше is used very often where English would say better:

• говорить лучше = to speak better
• понимать лучше = to understand better
• лучше чувствовать ритм = to feel the rhythm better

This is completely normal and idiomatic.

They are in the accusative case because they are direct objects of the verbs:

• чувствовать ритм = to feel the rhythm
• держать такт = to keep the beat / keep time

However, for masculine inanimate nouns, the accusative singular looks the same as the nominative singular. So:

• nominative: ритм, такт
• accusative: ритм, такт

That is why their forms do not change visibly here.

They are related, but not exactly the same.

• ритм = rhythm, the overall rhythmic pattern or sense of timing
• такт in music can mean beat, measure/bar, or the sense of keeping steady time

In the phrase держать такт, it usually means to keep time or maintain the beat steadily.

So the teacher is saying the violinist needs:

1. to feel the rhythm better, and
2. to keep steady time

These are close ideas, but not identical.

Yes, it is a fairly set musical expression.

Держать такт means to keep time or keep the beat. You should understand it as a natural collocation, not just literally hold the beat.

Russian often uses verbs in combinations that sound slightly unusual if translated word-for-word into English. This is one of those cases.

Because должен governs both infinitives.

The structure is:

• скрипач должен [лучше чувствовать ритм] и [держать такт]

English works similarly:

• The violinist should feel the rhythm better and keep time.

You do not need to repeat должен before the second infinitive, though you could in some contexts for emphasis.

They are imperfective because the sentence is talking about an ongoing skill or general requirement, not a single completed action.

• чувствовать ритм = to have a sense of rhythm
• держать такт = to keep time steadily

If Russian used perfective verbs here, it would sound more like one-time completed actions, which would not fit the idea of musical ability or performance technique.

So imperfective is the natural choice.

The word order is neutral and natural.

• Учительница сказала, что скрипач должен лучше чувствовать ритм и держать такт.

This is basically: The teacher said that the violinist should better feel the rhythm and keep time.

Russian word order is more flexible than English, but this version sounds straightforward and standard. Different word orders are possible for emphasis, for example putting лучше or ритм in a more prominent position, but the given order is the most neutral.

In standard Russian with сказать, что is normally used when introducing a full clause like this.

• Учительница сказала, что... = The teacher said that...

Leaving it out would usually sound ungrammatical or very nonstandard in this kind of sentence. So for learners, it is safest to think of сказать, что... as a regular pattern.

It can be translated either way depending on tone and context.

• must if the teacher sounds strict or the requirement is strong
• should if it is more like advice or criticism

In many teaching contexts, English should sounds more natural:

• The teacher said that the violinist should feel the rhythm better and keep time.

But grammatically, Russian должен itself is often stronger than English should, so context matters.