K.A.
We went through this when our 14 year old cat got ill and was at the end of his life before we had to put him to sleep. Our children were 7 & 5 at the time I think and I was pregnant with our third. We told them that his body was very old and could not work very well anymore. He was in a lot of pain. They went with us to the vet when we put him to sleep and they were in the room to say goodbye. The night before we put him to sleep he had lost the ability to walk so they knew it was coming. We explained that the vet was going to give him special medicine that would take the pain away so his body and mind could relax and not feel the pain so his spirit could leave the body that wasn't working and move on. We told them this is what dying means. The vet was also very good at telling what he was doing, he was amazingly comforting and careful. He'd been treating our cat the last few months of his life.
We were very careful not to use the word "Sleep" in refering to anything so the kids would not get scared that they or we could fall asleep and die.
There is a really good book by Mr Rogers about losing a pet that we bought that was very good that you may want to look into. http://www.amazon.com/When-Pet-Dies-Fred-Rogers/dp/069811...
So sorry for your loss.