Na talerzu została tylko resztka ciasta.

Because the grammatical subject is resztka (a small leftover bit / a remnant), and resztka is:

• feminine
• singular

So the past-tense verb must agree with it:

• została = feminine singular
• zostało = neuter singular
• zostały = non-masculine personal plural, etc.

Also, zostać here means to remain / to be left, not just to be.

So:

• Na talerzu była tylko resztka ciasta = There was only a bit of cake on the plate
• Na talerzu została tylko resztka ciasta = Only a bit of cake was left on the plate

The version with została emphasizes that this is what remained after the rest was gone.

In Polish, many nouns ending in -a are feminine, and resztka is one of them.

Dictionary form:

• resztka = a small remainder, a scrap, a leftover bit

Because it is feminine, the past tense is:

• została

If the subject were masculine, you would get:

• został

If it were neuter:

• zostało

So the ending of the verb is one of the main clues showing the gender of the subject.

Because na can take different cases depending on meaning.

Here it means on the plate in the sense of location, so it takes the locative case:

• talerz = plate
• na talerzu = on the plate

Compare:

• położyć coś na talerz = to put something onto the plate
  Here na expresses movement toward a place, so you use the accusative.
• coś jest na talerzu = something is on the plate
  Here na expresses location, so you use the locative.

So in this sentence, since the cake remnant is already sitting there, na talerzu is correct.

Because resztka is followed by a noun in the genitive.

This is very common in Polish with words meaning:

• a piece of something
• a bit of something
• a lot of something
• a remnant of something

So:

• ciasto = cake
• resztka ciasta = a remnant / leftover bit of cake

This is similar to:

• kawałek chleba = a piece of bread
• szklanka wody = a glass of water
• trochę czasu = a bit of time

So ciasta here is genitive singular, not nominative.

This is a very useful distinction.

• reszta = the rest / the remainder
• resztka = a small leftover piece / a scrap / a tiny remnant

So:

• Zjadłem ciasto, a reszta została na stole.
  = I ate the cake, and the rest stayed on the table.
• Na talerzu została tylko resztka ciasta.
  = Only a small leftover bit of cake remained on the plate.

In your sentence, resztka sounds natural because it suggests that very little was left.

Here tylko means only, and it highlights resztka ciasta as the thing that remained.

So the sentence means roughly:

• What was left on the plate? Only a bit of cake.

Polish word order is flexible, but placement changes emphasis. In this sentence:

• Na talerzu została tylko resztka ciasta.

the focus is on only a remnant of cake.

You may also hear:

• Tylko resztka ciasta została na talerzu.
• Na talerzu tylko resztka ciasta została.
  (less neutral, more marked)

The original version sounds natural and neutral.

Polish usually does not need a dummy subject like English there.

English often says:

• There is...
• There was...
• There remained...

Polish normally just uses the verb and the real noun:

• Na talerzu została tylko resztka ciasta.

Literally, it is closer to:

• On the plate remained only a remnant of cake.

This is completely normal in Polish. The sentence does not need any equivalent of English existential there.

Yes, the word order is normal, and yes, you can rearrange it.

The original:

• Na talerzu została tylko resztka ciasta.

This is a very natural order. It starts with the location, then gives the verb, then the subject phrase.

Other possible versions:

• Tylko resztka ciasta została na talerzu.
  Stronger focus on only a bit of cake
• Resztka ciasta została na talerzu.
  Neutral, but without tylko
• Na talerzu tylko resztka ciasta została.
  More marked, less standard in neutral speech

Polish word order is often used for emphasis and information structure, not just grammar.

Good question. Yes, zostać can mean to become, but not in this sentence.

Two common meanings of zostać are:

1. to remain / to be left
2. to become

Examples:

• Został w domu. = He stayed/remained at home.
• Została nauczycielką. = She became a teacher.

In your sentence:

• Na talerzu została tylko resztka ciasta.

it clearly means remained / was left, because the context is about what stayed on the plate.

Yes, you could say:

• Na talerzu pozostała tylko resztka ciasta.

This also means Only a bit of cake remained on the plate.

The difference is mostly stylistic:

• została = very common, everyday
• pozostała = a bit more formal or literary, but still normal

So both are correct, but została sounds very natural in everyday speech.

It is best understood as a noun phrase meaning:

• a leftover bit of cake
• a remnant of cake
• a small piece of cake left over

Structure:

• resztka = leftover bit, tiny remainder
• ciasta = of cake

So the sentence is not talking about the whole remaining cake, but just a small leftover amount.

That nuance is important: resztka suggests very little.

A close breakdown is:

• Na = on
• talerzu = plate (locative: on the plate)
• została = remained / was left
• tylko = only
• resztka = a small remnant / leftover bit
• ciasta = of cake

So literally:

• On the plate remained only a remnant of cake.

Natural English:

• Only a bit of cake was left on the plate.