Many people become attached to pets and don't want to have them "put to sleep" when they are old and sick. The whole business about Candy's dog was put into the story to establish that there was a gun available, so that George could steal it and use it later to kill Lennie. Carlson explains where to point the gun so that the dog's death will be painless, and so George knows where to point it when he shoots Lennie. Furthermore, Carlson cleans the Luger carefully after using it, and George is able to see exactly how the German automatic pistol works and where it is kept under Carlson's bunk. Carlson is a middle-aged man. We assume without being told that he served in the army in Europe during World War I and brought the Luger back as a souvenir. Steinbeck wanted a handgun that was readily recognizable, so that when George takes it out of his jacket to shoot Lennie, the reader (and eventually the theater audience when the novel is adapted into a stage play) will understand that it is Carlson's gun. A German Luger is a very distinctive-looking weapon. Steinbeck knew that every scene in a work of fiction should be dramatic. Drama involves conflict. He devised a conflict between Carlson and Candy over killing the dog in order to make the scene more interesting. The conflict is: Carlson is bothered by the old dog's smell and wants to shoot it; but Candy is attached to the dog and doesn't want to let Carlson shoot it. The conflict is resolved when Candy yields to pressure and the dog is shot. Notice the amount of explanation Steinbeck has given to the way Carlson unloads, cleans, and reloads his Luger after shooting the dog. The gun is what in Hollywood is called a "plant." A weapon is shown in advance, so that the audience won't wonder where it came from when it is used later in the story. If George had pulled a gun out of his pocket when he joined Lennie at the river, the reader/viewer would wonder where he got it if he didn't recognize it as Carlson's Luger. George intended to kill Lennie from the time he left the ranch. His possession of the stolen Luger proves his intent. The men thought Lennie had taken it when he fled. This was convenient for George, mainly because he could let everybody assume that he had taken it away from Lennie and shot him in self-defense, using this as an alibi to avoid prosecution for what was actually first-degree, premeditated murder.
The most relevant answer to your question is that Candy does not want to kill his dog because it is an extension of him. What I mean by this is that since the dog has no "real" use left Carlson wants to kill it; we, as readers, can see that Candy is already thinking that he himself does not have much more "useful" time left. This also goes back to the theme of "usefulness" in the text. As a secondary answer this event also points to the idea that there is to sentimentality on the ranch: once something has ceased to serve a purpose it is discarded. Candy’s dog has been his only companion for years and years and yet when it is recognized as not being able to serve a purpose or there is something else that can do a better job (i.e. Slim’s puppies) then it is discarded. Lastly, this foreshadows the end of the book and Lennie’s eventual death at the hand of George.
On the simplest level, Candy doesn't want to kill his dog because he loves it and remembers lots of great times they had together.
On a more symbolic level, Candy knows that shooting the dog would be like shooting himself. (His dog is old and weak; he is disabled. To agree to get rid of one would be like getting rid of the other.)
Candy does not want to kill his dog because that dog has been with him ever since he was young. He is also upset that they said his dog is old and useless and Candy is worried that it could be him that is old and useless.
Candy doesn't want to kill his dog because his dog has been with Candy for a long time and his dog has helped him a lot around the farm.
Candy also is upset that his dog (that is old and weak) is going to die when Candy is just as old and weak.