I personally think that dogs understand better than humans. We have 2 dogs from the local shelter and they are both so loving. They love life. Shawna, the Aussie, was full grown when we got her and would roll on her back if anyone raised their voice for anything, but now she takes on all the dogs and is the mother of the pack. Jazzy, the Dane, was only 3 months old when we got her, but you could tell she knew she had been abandoned and then saved, now she thinks she is our 120 pound lap dog. George, the hound mix, still thanks us, we got him because someone dumped him in our neighbors' back yard and they couldn't take him, so we did, and we couldn't be happier. Even Tigger, who my husband brought home because someone brought Tig to where my husband worked and told my husband that if someone didn't take this pup that they were going to take it home and shoot him. Tigger is totally attached even today, 2 years later. Sox is the only one from a litter, but 4 of the 6 pups died from worms because the people are irresponsible morons and even she seems to realize how lucky she is.
Enough blithering, my only point is that dogs know when they have been saved and are far more appreciative than any human could hope to be.