Odor fumi e culina venit, et mater statim servam vocat.

Because fumi is the genitive singular of fumus, meaning of smoke.

In odor fumi, Latin is literally saying the smell of smoke. English often uses of for this idea, and Latin commonly uses the genitive case.

So:

• fumus = smoke
• fumi = of smoke

That is why odor fumi means the smell of smoke.

It does change in sense, but here culina is already in the correct form.

The preposition e (or ex) takes the ablative case, and culina is a first-declension noun whose ablative singular is also culina.

So:

• nominative: culina = kitchen
• ablative singular: culina = from/out of the kitchen

Even though the form looks the same as the nominative, its job in the sentence is different.

Both e and ex mean out of or from and both take the ablative.

Usually:

• ex is used before vowels and often before some consonants
• e is used before many consonants

So e culina means from the kitchen. You may also see ex in other contexts. The choice is mostly about sound and style, not a difference in meaning here.

Because Latin word order is much freer than English word order.

English usually depends heavily on word order, but Latin usually shows relationships through case endings instead. That means Latin can move phrases around for emphasis or style.

So all of these would be understandable Latin with roughly the same basic meaning:

• Odor fumi e culina venit
• E culina odor fumi venit
• Odor e culina fumi venit (less natural stylistically, but still understandable)

In your sentence, venit comes at the end of the clause, which is very common in Latin.

Because servam is the accusative singular of serva, and it is the direct object of vocat.

The sentence says that the mother calls the slave-woman / maidservant, so Latin puts serva into the accusative:

• serva = a slave-woman, maidservant
• servam = the slave-woman, as the object

This is one of the most important things to notice in Latin: the ending shows what role the noun plays.

Because mater is the subject of vocat.

The person doing the calling is the mother, so Latin uses the nominative case:

• mater = mother, as subject
• matrem = mother, as direct object

In this sentence, mater is the one who acts, while servam receives the action.

Both are present tense.

• venit = comes
• vocat = calls

So the sentence describes actions happening now or in a vivid present-time narrative:

• Odor fumi e culina venit = the smell of smoke comes from the kitchen
• mater statim servam vocat = the mother immediately calls the maidservant

A learner should be careful with venit, because in other contexts the same spelling can also represent a perfect form from a different verb pattern, but here it is simply the present tense of venire.

Statim is an adverb meaning immediately, at once, or straightaway.

It modifies vocat, telling you when or how quickly the mother calls the servant.

So:

• mater vocat = the mother calls the servant
• mater statim servam vocat = the mother immediately calls the servant

Classical Latin has no definite article like English the, and no indefinite article like a/an either.

So:

• mater can mean mother or the mother
• servam can mean a slave-woman or the slave-woman
• culina can mean kitchen or the kitchen

The exact sense depends on the context. In a story, English translators usually add the where it sounds natural.

Yes. Serva is feminine and means female slave or sometimes maidservant, depending on context.

Compare:

• servus = male slave, slave-man
• serva = female slave, slave-woman, maidservant

Since your sentence has servam, it is clearly referring to a female servant.

Yes. Even though the Latin structure is literally smell of smoke, natural English might say:

• the smell of smoke
• a smell of smoke
• the smoky smell
• the smell of smoke comes from the kitchen

The grammar point, though, is that Latin uses the genitive in odor fumi to express the of smoke relationship.

Et means and, simply linking the two clauses:

• Odor fumi e culina venit
• et mater statim servam vocat

Latin often joins ideas with et where English might also use and, or sometimes split them into separate sentences. Here it shows that the second action follows naturally from the first: smoke smell comes from the kitchen, and the mother immediately calls the maidservant.