Kahvaltıda tereyağı ve kaymak vardı; ben ekmeğe ikisini de sürdüm.

Yes, literally it is kahvaltı + -da, the locative ending.

• kahvaltı = breakfast
• -da/-de = in, at, on, during, depending on context

With meal names, Turkish often uses the locative to mean at breakfast, at dinner, during lunch, etc. So Kahvaltıda is best understood as At breakfast or For breakfast, not as a physical location.

Var means there is / there are / exists.
Its past form vardı means there was / there were.

So:

• tereyağı ve kaymak vardı = there was butter and kaymak
• more natural English: there were butter and kaymak available / on the table

Turkish uses var and yok very often for existence or availability. In this sentence, the idea is not just that butter and kaymak were in some abstract sense, but that they were there at breakfast.

Yes. Turkish normally does not need the subject pronoun, because the verb ending already shows the person.

• sürdüm = I spread

So ben is optional here. When it is included, it usually adds emphasis or contrast, something like:

• I spread both of them
• as for me, I spread both of them

It can sound slightly more deliberate than just ekmeğe ikisini de sürdüm.

Tereyağı is a compound noun meaning butter. In Turkish, many compound nouns include a built-in ending like this as part of the dictionary form.

So you should learn tereyağı as the whole word, not as tereyağ plus some extra ending added only here.

A similar example is:

• zeytinyağı = olive oil

When case endings are added, they attach to the whole compound:

• tereyağı = butter
• tereyağını = the butter (accusative)
• tereyağına = to the butter

This is a normal sound change in Turkish.

The base word is:

• ekmek = bread

When a vowel-initial ending is added, final k often softens to ğ. So:

• ekmek
  • -e (dative) → ekmeğe

This kind of consonant softening is very common in Turkish.

Because with sürmek in the sense of to spread/apply something onto something, Turkish often uses:

• the thing being spread = direct object
• the surface it is spread onto = dative

So here:

• ikisini de = both of them (the things being spread)
• ekmeğe = onto the bread

This is why ekmeğe sürdüm means I spread it onto the bread.

If you used ekmeği instead, that would make bread the direct object, which would change the structure and not fit this meaning.

Because ikisini is the accusative form, and here it is the direct object of sürdüm.

Compare:

• ikisi de = both of them
  This is usually used as a subject or topic.
• ikisini de = both of them
  This is used when they are the object of the verb.

In this sentence, both of them are what the speaker spread, so the object form is needed:

• ikisini de sürdüm = I spread both of them

Here de adds the meaning of too / also / as well, but in combinations like ikisini de, it helps give the sense of both.

So:

• ikisini de = both of them
• literally something like the two of them as well

It is written separately because this de is a separate particle, not a suffix.

This is different from the locative ending -de/-da, which is attached to the word:

• kahvaltıda → attached suffix
• ikisini de → separate particle

That spelling difference is very important in Turkish.

It can. Sürmek is a very flexible verb in Turkish, and its meaning depends a lot on context.

Common meanings include:

• to drive a car
• to drag
• to continue/last
• to spread/apply/rub

Here, because the sentence is about butter and kaymak on bread, sürmek clearly means to spread.

Formally:

• sür- = verb root
• -dü- = past tense
• -m = first person singular

So sürdüm = I spread.

Because Turkish does not have a definite article like English the.

Definiteness is usually understood from:

• context
• word order
• case marking
• whether the noun is already known in the conversation

So ekmeğe can naturally be translated as onto the bread, even though there is no separate word meaning the.

In this sentence, the breakfast setting makes the bread and the foods feel contextually identifiable, so English often uses the, while Turkish does not need to.

Turkish word order is fairly flexible, as long as the case endings make the roles clear.

This sentence is:

• ben ekmeğe ikisini de sürdüm

But you could also hear:

• Ben ikisini de ekmeğe sürdüm
• İkisini de ekmeğe sürdüm

The basic meaning stays the same. The differences are mostly about emphasis and flow.

The original version feels natural because it presents:

1. the setting and available foods
2. then the speaker’s action
3. with the verb at the end, which is very typical in Turkish