Yemeği soğutucu çantaya koyduk.

Because yemeği is yemek + the accusative ending.

• yemek = food / a meal
• yemeği = the food / the meal

In Turkish, a specific direct object often takes the accusative ending. Here, the speaker is talking about a particular food, not just food in general.

There is also a sound change:

• yemek
  • -i → yemeği

The final k often changes to ğ when a vowel-initial suffix is added.

Because çantaya has the dative ending, which here gives the idea of to / into the bag.

• çanta = bag
• çantaya = to the bag / into the bag

The ending is -(y)A:

• after a vowel, Turkish often adds a buffer y
• çanta
  • -ya → çantaya

In this sentence, with koymak (to put), the dative often marks the place something is put into or onto.

Soğutucu çanta means something like cooler bag, insulated bag, or cooling bag.

• soğutucu comes from a verb related to cooling
• here it functions like an adjective describing çanta

So:

• çanta = bag
• soğutucu çanta = cooler bag

This is very normal in Turkish: a descriptive word comes before the noun, just like in English.

Koyduk can be broken down like this:

• koy- = put / place
• -du- = past tense
• -k = we

So:

• koyduk = we put

This is one of the most useful things about Turkish verbs: the verb ending already tells you the subject, so you do not need to say biz (we) unless you want emphasis.

Because Turkish usually does not need subject pronouns when the verb ending already makes the subject clear.

• koydum = I put
• koydun = you put
• koydu = he/she/it put
• koyduk = we put

So koyduk already includes the meaning we.

If you said Biz yemeği soğutucu çantaya koyduk, it would still be correct, but biz would add emphasis, like We put the food in the cooler bag.

Because Turkish is usually an SOV language:

• Subject – Object – Verb

So a very typical Turkish sentence places the main verb last.

Here:

• Yemeği = the food
• soğutucu çantaya = into the cooler bag
• koyduk = we put

Literally, the structure is closer to:

• The food into the cooler bag put-we

That sounds odd in English, but it is normal in Turkish.

Yes, Turkish word order is somewhat flexible, but the verb usually stays at the end.

For example, these can all be possible depending on emphasis:

• Yemeği soğutucu çantaya koyduk.
• Soğutucu çantaya yemeği koyduk.

The version you were given is natural and neutral. Changing the order usually changes focus or emphasis, not the core meaning.

The difference is mainly specificity.

• yemek koyduk = we put food / we put some food
• yemeği koyduk = we put the food / we put the meal

So the accusative ending on yemeği shows that the speaker means a specific, known thing.

This is a very important Turkish pattern:

• bare noun = often more general or indefinite
• accusative-marked noun = specific definite object

It can mean either food or meal, depending on context.

So yemeği could be understood as:

• the food
• the meal

If the broader situation is about packing lunch, it might feel more like the meal. If it is about placing food items in a cooler bag, it might feel more like the food.

Both are possible.

Because they are different case endings with different jobs.

• -i here is the accusative: marks the specific direct object
• -ya here is the dative: marks direction, often to / into

So in this sentence:

• yemeği = the thing being put
• çantaya = the place it is being put into

They are not interchangeable, because they show different roles in the sentence.