take care of the dog犬の面倒を見る means “to take care of the dog.” The て-form can connect to another clause or be used in a request, such as 犬の面倒を見てくれる？ (“Will you take care of the dog?”).
JA: 旅行中、犬の面倒を見てくれない？
EN: Could you take care of my dog while I’m traveling?