Например: детям дали по сырнику, а взрослым — по чашке кофе.

Because they are the recipients of дали (gave).

The verb дать / давать typically takes:

• the person receiving something in the dative
• the thing being given as the direct object

So here:

• детям = to the children
• взрослым = to the adults

Compare:

• Я дал брату книгу. = I gave my brother a book.
• Им дали подарок. = They were given a present.

In your sentence, the recipients are the children and the adults, so both are in the dative.

Here по has a distributive meaning: one each / apiece.

So:

• детям дали по сырнику = the children were given one syrnik each
• взрослым — по чашке кофе = the adults got one cup of coffee each

This use of по is very common in Russian when something is distributed individually:

• дали по яблоку = gave one apple each
• выдали по билету = handed out one ticket each
• купили по чашке чая = bought a cup of tea each

Without по, the sentence would not so clearly express the idea of each person getting one.

Because after distributive по, Russian often uses the dative singular when the meaning is one each.

So:

• по сырнику = one syrnik each
• по чашке кофе = one cup of coffee each

That is why you see:

• сырник → сырнику
• чашка → чашке

This is a very standard pattern:

• по карандашу = one pencil each
• по книге = one book each
• по бутерброду = one sandwich each

For an English speaker, this can feel strange because English uses each as a separate word, while Russian often expresses that meaning with по + dative.

The dash marks an omitted word or words, usually because they are easy to understand from the previous part.

The full version would be:

• Например: детям дали по сырнику, а взрослым дали по чашке кофе.

In the second part, дали is left out because it would just repeat what was already said. The dash shows that omission.

This kind of ellipsis is very common in Russian, especially in comparisons and contrasts:

• Детям — сок, взрослым — чай.
• Одним дали книги, другим — тетради.

So the dash here is natural and elegant, not unusual.

Because Russian often omits the subject when it is unknown, unimportant, or understood from context.

дали is the past plural form of дать, literally they gave. But in many sentences, it works like an indefinite subject:

• дали = they gave, someone gave, or even were given depending on translation

So this sentence does not have to mean that a specific group of people is being emphasized. It simply reports the action:

• детям дали по сырнику = the children were given a syrnik each

This is a common Russian way to express actions without naming the doer.

А often marks a contrast or comparison between two parts of a sentence.

Here the sentence is contrasting what two groups received:

• the children got one thing
• the adults got another

So а works very naturally:

• детям дали по сырнику, а взрослым — по чашке кофе

This is not a strong contradiction, so но (but) would sound too strong in many contexts.

And и just means and, without highlighting the contrast as clearly.

A good rough sense of а here is:

• whereas
• while
• and as for

Because кофе is an indeclinable noun in standard Russian. That means its form stays the same in all cases.

So even though the phrase чашке кофе involves a case relationship, кофе still looks like кофе.

Compare with a declinable noun:

• чашка чая = a cup of tea
• по чашке чая = a cup of tea each

Here чай changes to чая.

But with кофе:

• чашка кофе
• по чашке кофе

same form both times.

In this phrase, кофе functions like the substance contained in the cup: a cup of coffee.

Because дали is perfective, while давали is imperfective.

• дали = gave, in the sense of a completed act
• давали = were giving / used to give / gave repeatedly

In this sentence, the idea is that the items were handed out as a completed event, so дали is the natural choice.

Compare:

• На празднике детям дали по сырнику. = At the celebration, the children were given a syrnik each.
  (one completed action)
• Каждое утро детям давали молоко. = Every morning the children were given milk.
  (repeated action)

A сырник is a traditional dish made mainly from творог (farmer's cheese / curd cheese), usually shaped into small patties and fried.

English has no perfect one-word equivalent, so learners often see translations like:

• cheese pancake
• curd-cheese fritter
• syrnik (left untranslated)

So in this sentence, по сырнику means each child got one of those.

Because например is introducing an example, and the colon signals that what follows is that example.

So the structure is:

• Например: = For example:
• then the example itself

This punctuation is normal in Russian, especially when например introduces a full example rather than being inserted inside a sentence.

You could also see например used without a colon in other sentence structures, but here the colon is completely natural.