После ужина остались дыня и персики, поэтому утром был лёгкий завтрак.

Because после requires the genitive case.

• ужин = dinner
• после ужина = after dinner

This is a very common pattern in Russian:

• после работы = after work
• после урока = after the lesson
• после дождя = after the rain

So in this sentence, ужина is the genitive singular form of ужин.

Because its subject is дыня и персики — two things together:

• дыня = melon
• персики = peaches

A compound subject joined by и usually takes a plural verb in Russian, just like in English:

• дыня и персики остались = the melon and the peaches were left

So остались is the plural past-tense form of остаться.

Because остаться means not simply to be, but to remain / to be left over.

So:

• были would only mean were
• остались means remained or were left

That matches the idea that some food was still there after dinner.

Compare:

• На столе были фрукты. = There were fruit on the table.
• После ужина остались фрукты. = After dinner, some fruit was left.

Остались is the past tense plural form of остаться.

Breakdown:

• infinitive: остаться
• past masculine singular: остался
• past feminine singular: осталась
• past neuter singular: осталось
• past plural: остались

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

Here the subject is plural, so we get остались.

Because the sentence is talking about:

• one melon: дыня
• several peaches: персики

Russian often leaves number exactly as intended, just like English. So this combination is completely natural:

• дыня и персики

If there were several melons too, it could be:

• дыни и персики

But here it is one melon and multiple peaches.

Утром is the instrumental case form of утро, and in expressions of time it often means in the morning.

So:

• утро = morning
• утром = in the morning

This is a very common adverbial use of the instrumental:

• днём = in the daytime
• вечером = in the evening
• ночью = at night

So поэтому утром means therefore / so in the morning.

Because лёгкий завтрак is the subject, not a predicate noun in the instrumental.

In this sentence, the structure is basically:

• утром был лёгкий завтрак
• literally: in the morning, there was a light breakfast

Here:

• завтрак is nominative singular
• лёгкий agrees with it in nominative masculine singular

Russian often uses this kind of structure to say that some event or meal took place / was had.

If you said был лёгким завтраком, that would suggest something like was a light breakfast in a different grammatical relationship, which does not fit here.

Because был belongs to the second clause and agrees with завтрак, not with дыня и персики.

The sentence has two parts:

1. После ужина остались дыня и персики
2. поэтому утром был лёгкий завтрак

In the second part, the subject is:

• завтрак = breakfast

Since завтрак is masculine singular, the verb is:

• был

So:

• остались agrees with дыня и персики
• был agrees with завтрак

Лёгкий is an adjective modifying завтрак.

Because завтрак is:

• masculine
• singular
• nominative

the adjective must match it:

• лёгкий завтрак

Agreement in Russian adjectives depends on gender, number, and case.

Compare:

• лёгкий завтрак = a light breakfast
• лёгкая еда = light food
• лёгкое блюдо = a light dish
• лёгкие закуски = light snacks

It usually means a light breakfast in the sense of not heavy, not substantial, or not too filling.

In this context, it suggests that breakfast was simple, probably because the leftover fruit was eaten in the morning.

So лёгкий here is not really about weight. It is about the amount or heaviness of the meal.

Both are possible, but the chosen word order sounds natural because it sets the scene first:

• После ужина = after dinner

Then it introduces what remained:

• остались дыня и персики

Russian word order is flexible, and moving elements changes emphasis more than basic meaning.

Very roughly:

• После ужина остались дыня и персики
  focuses first on the situation after dinner
• Дыня и персики остались после ужина
  puts more emphasis on the melon and peaches

The original order is smooth and natural in narrative style.

Поэтому means therefore, so, or that’s why. It connects the two clauses logically:

• fruit was left after dinner
• therefore, breakfast in the morning was light

Its position is fairly flexible, but placing it near the start of the second clause is very common:

• ..., поэтому утром был лёгкий завтрак.

Other placements can be possible in different contexts, but this is the most straightforward and natural one.

Russian often expresses there was / there were simply with быть in the past tense, without adding a separate word for there.

So:

• был лёгкий завтрак literally looks like
• was a light breakfast

but naturally it corresponds to:

• there was a light breakfast or
• breakfast was light / there was a light breakfast

Russian does not need a dummy subject like English there.

Yes, остались comes from the perfective verb остаться.

That makes sense because the sentence refers to a resulting situation:

• after dinner, some fruit ended up remaining

Perfective often highlights the result or completed fact.

In contrast, the second verb был is from быть, which works differently and does not present the same perfective/imperfective contrast in this sentence.

So the main aspect point here is that остались presents the leftover fruit as a completed result of dinner.