Literally, the reason that Lennie likes these kinds of small animals is that he likes to pet them. In Chapter 1, that is what he talks about wanting to do. We also see this because his Aunt Clara gave him a rubber mouse he did not like it because it was no good to pet.
I think that figuratively, this is meant to show how childish Lennie is and how much he depends on George. He is so obsessed with little things that he can pet that he seems like a child.
Lennie is not only described as a person who likes animals (like rabbits, mice and puppies). He is also characterized as being animalistic (or animal-like). Lennie is repeatedly connected to the wild, natural world and shows that he can act instinctively and violently, as animals do. We might argue that Lennie is incapable of becoming truly socialized and so "belongs" to the animal world in some ways.
This take on Lennie's character is not meant to be a harsh reading of a man that clearly has developmental challenges. Rather, it is simply a straight-forward way to answer questions as to (1) the nature of Lennie's connection to animal comforts (e.g. "soft things") and (2) the significance of the details associated with his character.
Consider Lennie's introduction in the text.
"[...] he walked heavily, dragging his feet a little, the way a bear drags his paws."
In the following paragraph when Lennie reaches the brook he "drank with long gulps, snorting into the water like a horse."
These specific details are reiterated and echoed again at the end of the novel when Lennie runs back to the brook after killing Curley's wife. There again he is depicted as an animal and compared to a bear.
Lennie's connection to mice and puppies is explained in part via this metaphorical, animalistic characterization. Lennie has an affinity for animals because, in a figurative way, he has a kinship with them.
What is the significance of this kinship in the text? We might answer this question in a number of ways and you may have your own ideas about why Lennie's connection to animals (and a sort animalism) is important.
To briefly offer some ideas on this, we can read Of Mice and Men as a story of man's break from nature and/or read the book as a commentary on certain specific challenges placed on people living in a society where friendship is pitted against property ownership (and economic concerns).
Reading Of Mice and Men as a story about a break from nature, we will pay close attention to the opening and closing sections of the work, noting how Lennie is characterized, how George and Lennie find peace in nature and how Lennie also threatens to run off to live in the wild. These details can be connected to an interpretation of the book wherein Lennie is a person who cannot successfully make the break from nature and who is incapable also of subduing his own inner-nature. He has no place in the society of the novel and no capacity to discover one.
"That Lennie has to die at the novel's conclusion is a poignant commentary on the inability of the innocent to survive in modern society" (eNotes).
Such a conclusion also aligns with a more Marxist interpretation of Steinbeck's work. Reading Of Mice and Men as a commentary on economic pressure as a stress on human relationships, we can look at the way friendships are often organized around the dream of property ownership. Lennie's love for soft things leads him to pet Curley's wife's hair then leads him to kill her as a way of protecting his one chance at continued friendship - the dream of the ranch that he and George often talk about. With this dream lost, the bonds that cemented the friendships between George and Lennie and Candy are (dramatically) severed.
Lennie likes to pet soft things the animals fur is soft and he enjoys petting them. He is rough and sometimes gets carried away with the feeling of the soft animal fur therefore harming the animals and sometimes killing them. This is accidental he dose not want to harm anything and feels the guilt and pain when he realizes he has hurt a living thing.