Learn Turkish: Yatağı söktükten sonra odaya yerleştirdik. - After dismantling the bed, we placed it in the room.

It takes the accusative case suffix -ı because the bed is the specific, defined object being dismantled. When you add a vowel suffix to a word ending in -k, that consonant softens into a -ğ (yatak -> yatağı).

How does söktükten sonra translate to "after dismantling"?

To say "after doing [action]" in Turkish, you take the verb root (sök-) and add the past participle suffix -DIk (which becomes -tük due to vowel harmony and consonant assimilation), followed by the ablative "from" case -tAn (becoming -ten), and finally the word sonra. Literally, it means "after from the dismantling".

The English translation says "we placed it", but I don't see the word for "it" (onu). Why?

In Turkish, it is very common to drop pronoun objects like onu (it/him/her) if the object is already obvious from the context. Since yatağı (the bed) was just established as the object in the first half of the sentence, repeating it with onu is unnecessary.

Why does oda (room) take the dative case (odaya)?