Мама сказала, что для теста нужны дрожжи, иначе корж будет слишком тяжёлым.

Because мама is grammatically feminine, and the Russian past tense agrees with the subject in gender and number.

• мама сказала = Mom said
• папа сказал = Dad said

In the past tense, Russian verbs often show:

• masculine: сказал
• feminine: сказала
• neuter: сказало
• plural: сказали

Что introduces a subordinate clause and means that:

• Мама сказала, что... = Mom said that...

In English, that is often optional, but in Russian что is very commonly used in this kind of sentence.

Because the preposition для always requires the genitive case.

The dictionary form is тесто = dough / batter.
Its genitive singular form is теста.

So:

• тесто = dough, batter
• для теста = for the dough / for the batter

This is also a useful warning: теста here is not related to English test. It is just the genitive form of тесто.

Because it agrees with дрожжи, which is grammatically plural.

• дрожжи нужны = yeast is needed / yeast is necessary

Even though English usually treats yeast as uncountable/singular, Russian дрожжи is a plural-only noun.

So Russian uses plural agreement:

• дрожжи нужны
• not дрожжи нужен

This is just a difference between the two languages. Russian дрожжи is a pluralia tantum noun, meaning it exists only in plural forms.

Other Russian nouns work this way too. So even when English uses a singular or uncountable noun, Russian may use a plural form.

That means you get plural agreement:

• дрожжи свежие = the yeast is fresh
• дрожжи были = the yeast was
• дрожжи нужны = the yeast is needed

Not exactly. Нужны is the short-form adjective of нужный and means needed / necessary in this sentence.

Russian often uses these short forms predicatively:

• Мне нужен сахар. = I need sugar. / Sugar is needed for me.
• Нам нужны дрожжи. = We need yeast.

In your sentence:

• для теста нужны дрожжи = yeast is needed for the dough

There is no present-tense form of to be written here. Russian usually omits есть in the present tense.

Both are possible. Russian word order is flexible and often depends on emphasis.

• Для теста нужны дрожжи puts early focus on for the dough
• Дрожжи нужны для теста puts early focus on yeast

The version in the sentence sounds natural because it first sets the context (for the dough) and then says what is needed.

Иначе means otherwise / or else.

Here:

• иначе корж будет слишком тяжёлым = otherwise the cake layer/crust will be too heavy

It often appears near the beginning of a clause, but Russian word order is somewhat flexible. Starting the clause with иначе is very common because it clearly signals a consequence.

Корж is a baking word. Depending on context, it can mean:

• a cake layer
• a cake base
• a sponge layer
• sometimes a crust

In this sentence, it most likely means the baked cake layer or base that would turn out too dense/heavy without yeast.

After forms of быть (to be), Russian very often uses the instrumental case for a predicate noun or adjective, especially in past and future contexts.

So:

• корж будет тяжёлым = the cake layer will be heavy

Here тяжёлым is the instrumental singular masculine form of тяжёлый.

This is the pattern to notice:

• Он был уставшим. = He was tired.
• Она будет готовой. = She will be ready.
• Корж будет слишком тяжёлым. = The cake layer will be too heavy.

Слишком means too in the sense of excessively.

• слишком тяжёлым = too heavy

It normally goes directly before the adjective or adverb it modifies:

• слишком большой = too big
• слишком медленно = too slowly
• слишком тяжёлым = too heavy

So its position here is exactly what you would expect.

Because Russian often states the actual noun instead of using a placeholder pronoun like English it.

English might say:

• otherwise it will be too heavy

Russian simply says:

• иначе корж будет слишком тяжёлым
• literally: otherwise the cake layer will be too heavy

That is completely natural in Russian.