This is the 11723rd most frequent Turkish word.
gayretlerine
"To their efforts" or "to their endeavors."
In this sentence, 'gayretlerine' is used in a dative sense to refer to efforts that belong to 'his'. It is joined with the preposition 'to' (related to) to indicate the direction of admiration.
Onun gayretlerine hayran kaldık.
We admired his efforts.
Here, 'gayretlerine' connects to the action of thanking, showing that gratitude is directed towards his efforts.
Gayretlerine teşekkür etmeliyiz.
We should thank him for his efforts.
The word 'gayretlerine' indicates the efforts that deserved reciprocation, modifying the action of awarding as a response to these efforts.