As he promised, George is able to get Lennie a puppy from the litter that has just been born to Slim's dog. It is detailed throughout the story that Lennie has an obsession with petting soft things. In chapter one, after taking away the dead mouse, George tells Lennie that he will try to find a puppy for him:
“Tell you what I’ll do, Lennie. First chance I get I’ll give you a pup. Maybe you wouldn’t kill it. That’d be better than mice. And you could pet it harder.”
In chapter two, George and Lennie are introduced to several characters including Candy, Slim and Carlson. During conversation, Carlson suggests that Slim give one of his new puppies to Candy to replace the old swamper's dog which is old and decrepit and needs to be put down. Lennie hears this information and begs George to ask Slim for one of the puppies. At the beginning of chapter three as Slim and George enter the bunkhouse, the reader learns that Slim has indeed given one of his puppies to Lennie. George expresses his gratitude and claims that Lennie will want to spend all of his time with the new puppy and will probably want to sleep in the barn with the dogs. George says,
“It wasn’t much to you, maybe, but it was a hell of a lot to him. Jesus Christ, I don’t know how we’re gonna get him to sleep in here. He’ll want to sleep right out in the barn with ‘em. We’ll have trouble keepin’ him from getting right in the box with them pups.”
Because of Lennie's fascination with soft things such as the dead mouse and Curley's wife's hair, he wants to go to the barn and pet the puppies. It appears that soft things are mentally soothing to Lennie. They give him a sense belonging in a world where he is considered an outcast.
You are referring to the part of the story that occurs after the litter of puppies is born. Lennie has a fascination for small and furry animals and really treasures them and the feel of their fur on his skin. This is demonstrated in Chapter 1 when George accurately guesses that Lennie has been carrying a mouse in his pocket. Lennie will want to sleep in the barn because Lennie will want to be near the puppies.