Learn Turkish: Beyefendi şapkasını vestiyere asıyor. - The gentleman is hanging his hat in the cloakroom.

It breaks down as şapka (hat) + -sı (his, 3rd person possessive) + -n- (buffer letter) + -ı (accusative case). The accusative case is needed because "his hat" is the specific direct object being acted upon by the verb asmak (to hang).

The English translation says "in the cloakroom", so why does Turkish use the dative vestiyere (to the cloakroom) instead of the locative vestiyerde?

In Turkish, verbs that involve placing, moving, or adding something to a location take the dative case (showing destination) rather than the locative (which shows static location). Since he is moving the hat onto a hook or shelf in the cloakroom, the verb asmak (to hang) requires the dative suffix -e.

How could we use the word ait (belonging to) from this lesson to talk about this hat?

If you wanted to say "The hat belongs to the gentleman", you would attach the dative case to the owner, followed by the postposition ait. The sentence would be: Şapka beyefendiye ait.

What kind of word is vestiyer?

Vestiyer is a loanword from French (vestiaire) meaning a cloakroom, coat check, or coat rack. It specifically refers to the room or piece of furniture near an entrance where outerwear and hats are left.