На одном шампуре были овощи, а на другом — кусочки мяса для шашлыка.

Because Russian treats a skewer as a surface/support that food is on: на шампуре = on a skewer.

A useful contrast:

• на шампуре = on the skewer / on a skewer
• на шампур = onto the skewer / onto a skewer

So in this sentence, the food is already there, which is why the location form на шампуре is used.

Шампуре is in the prepositional case singular.

That is because на takes the prepositional case when it means in/on/at a location:

• на шампуре = on a skewer

The word одном is the prepositional singular form of один and agrees with шампуре:

• на одном шампуре = on one skewer

So the whole phrase is built by agreement:

• на
  • одном
    • шампуре

Because Russian often leaves out a repeated noun when it is obvious from context.

So:

• на другом

really means:

• на другом шампуре

This is very natural. In English we do something similar with on the other one.

Here другом is the prepositional singular form of другой, agreeing with the omitted masculine noun шампур.

The dash marks an omission: the verb были is understood but not repeated.

Full version:

• На одном шампуре были овощи, а на другом были кусочки мяса для шашлыка.

Shorter, more natural version:

• На одном шампуре были овощи, а на другом — кусочки мяса для шашлыка.

So the dash is helping show that Russian has skipped over something already clear from the first clause.

Because it agrees with a plural subject.

In the first clause, the subject is овощи (vegetables), which is plural, so the past-tense verb is plural too:

• овощи были

In the second clause, the omitted verb would also be были, because кусочки is also plural:

• кусочки были

Russian past tense agrees in number, and in the singular it also agrees in gender.

Because а often expresses contrast or comparison, not just simple addition.

Here the structure is:

• on one skewer ... and on the other ...

That kind of contrast is exactly where а is natural.

Compare:

• и = and in a simple adding sense
• а = while / whereas / and as for the other one

So а fits the one-versus-the-other contrast much better.

Шампур means a skewer, especially a metal skewer used for grilling meat or vegetables.

It is a very common word in contexts involving шашлык.

Useful form here:

• dictionary form: шампур
• in the sentence: на шампуре

Also useful to know: the stress is шампУр.

Yes. Кусочки is the plural of кусочек, which is a diminutive form of кусок.

Very roughly:

• кусок = a piece / a chunk
• кусочек = a little piece / a small piece
• кусочки = small pieces

In this sentence, it does not necessarily sound cute or emotional. Very often the diminutive just means smaller pieces in a practical sense.

So кусочки мяса is naturally understood as pieces/chunks of meat.

Because after nouns meaning a piece, part, or amount, Russian usually puts the following noun in the genitive.

So:

• кусочек сыра = a piece of cheese
• кусок хлеба = a piece of bread
• кусочки мяса = pieces of meat

Here мясо changes to мяса, which is genitive singular.

That is especially natural because meat is a mass noun. Russian treats it as pieces of meat, not as separate countable meats.

Because the preposition для always takes the genitive case.

So:

• шашлык → шашлыка

That gives:

• для шашлыка = for shashlik

In context, this means the meat pieces are intended for making шашлык—that is, meat prepared for grilling on skewers.

In this sentence, it most naturally means the other one.

The pattern:

• на одном ... на другом ...

usually corresponds to:

• on one ... on the other ...

That said, другой can mean both other and another in different contexts. Here, because the sentence sets up a pair, the other is the best interpretation.

Russian word order is flexible, and this sentence puts the location first to highlight the contrast:

• На одном шампуре ... а на другом ...

That makes the two skewers the main organizing idea of the sentence.

A different order is also possible:

• Овощи были на одном шампуре, а кусочки мяса для шашлыка — на другом.

But the original version sounds very natural and does a good job of emphasizing one skewer vs. the other.