So sorry about your dog, they are part of the family. We put ours down last summer (12 1/2 years old) She was our "practice" child and a fierce protector of our kids and our home. My children were 9 and 11 at the time. WE got advice from friends who are counselors. Here are a few tips, but use your own words and whats appropriate for your child's age
1. Be sure to emphasize that "Fido's doctor says.... You want to make sure they differentiate their doctor and the dogs doctor. What we said is that the dog was very old and lived a very full life and her body parts just weren't working anymore and she is in a lot of pain. She was a wonderful friend to us and the kindest thing we can do for her now is help her by giving her medicine to take away the pain. For our children we added that this would make her heart stop. We also prefaced all of this by telling them that she was dying and in a lot of pain
2. Let them say goodbye but not at the vet. That would be too hard and make them afraid of getting shots in the future. We brought our dog home from the vet and all sat around her and said goodbye and told our favorite stories. Then my husband and I took her back to be put down. It was VERY hard for us and would have been too hard for our kids, even at their age. (plus will most likely make them terrified of shots)
3. Friends gave us many books,"Dog Heaven" was their favorite. We have her ashes and collar on our mantel. We talk of her often and after a year are almost ready to get a new dog
Hope this helps - good luck