If you feel that your pet is in danger, do whatever you can to shield him or her from harm—for instance, bring your outdoor cat inside and always accompany your dog outside, keeping him on a leash at all times.
You should also file a complaint with your local police; depending on the law where you live, verbal or written threats may constitute criminal harassment. Be sure to keep any tangible evidence of threats against your pets and yourself. If the threats are serious enough, you may be able to get a restraining order against the person making them. If this is the route you wish to go, enlist the aid of a lawyer. Above all, please be careful.