For many pet owners, the loss of a pet is an extremely sad time, and a gesture of condolence and support is appreciated. A comforting sympathy note or phone call offering a shoulder to cry on are both appropriate ways to express condolences.
The loss of a beloved pet leaves a void for the family, as one of their members is gone. While it may be said with the best of intentions, comments about the pet being in a better place or about the family getting another pet are not appropriate and are likely to cause more upset for those who are grieving. Simply acknowledging the loss and being available to talk provides condolences and support in an appropriate way.