How does the narrator in Sredni Vashtar encourage the reader to empathize with the protagonist?

There are several traits that Conradin himself possesses which are cause for the reader's sympathy. There are also other characteristics of the cousin/guardian which provide a reason for the reader to feel empathy or, at least, sad concern for Conradin in his misfortune.

To Conradin, his guardian named Mrs. De Ropp, 

represented those three-fifths of the world that are necessary and disagreeable and real; thwarting him "for his good" was a duty which she did not find particularly irksome. 

Mrs. De Ropp seems determined to prevent Conradin's happiness as she "thwarts him for his good." For instance, when the cousin notices that Conradin makes numerous trips to a forgotten old shed hidden by an overgrowth of shrubbery, she investigates by entering the shed where she discovers "a ragged-plumaged Houdan hen."

At this point in the narrative, the reader's sympathies are aroused with the discovery that there is a distinctively pretty Houdan hen, "on which the boy lavished an affection that had scarcely another outlet." Without friends with whom he can play, this hen provides Conradin an outlet for his love. Unfortunately for the boy, his cousin has noticed that he frequents the old shed. So, the cruel Mrs. De Ropp searches this shed and discovers the lovely pet. She then decides that it is not good for the boy to be "pottering down there (in the shed) in all weathers." Therefore, she announces the next morning that the Houdan hen has been removed and taken away.

Devastated by the loss of his pet, Conradin continues to go to the tool shed. But now his trips are for another reason:

Every Thursday, in the dim and musty silence of the tool-shed, he worshiped with mystic and elaborate ceremony before the wooden hutch where dwelt Sredni Vashtar, the great ferret.

Mrs. De Ropp has not noticed this animal. Even when removing the Houdan hen, she has not seen the ferret because it has always been hidden in the recesses of the old shed. Conradin prays secretly to this little god that he has fashioned in his imagination. He calls the ferret Sredni Vashtar and only asks him, "Do one thing for me," sobbing as he looks in the corner where his hen used to be. Then, Conradin returns "to the world he so hated." Each night he prays to Sredni Vashtar.

Of course, the cousin notices Conradin's trips to the shed have not stopped with the removal of the hen. Suspecting that the boy has some other animal hidden in the shed, she confronts Conradin:

"What are you keeping in that locked hutch?" she asked. "I believe it's guinea-pigs. I'll have them all cleared away."

Conradin tells her nothing. His guardian ransacks his bedroom and finds a key. With this key, Mrs. DeRopp goes to the shed to search for what it is that the boy has hidden: 

He knew that the Woman would come out presently with that pursed smile he loathed so well on her face, and that in an hour or two the gardener would carry away his wonderful god, a god no longer, but a simple brown ferret in a hutch.

Because the lonely Conradin may be deprived of the last creature that he can enjoy and talk to, the reader's feelings for him become stronger, and the antipathy for the cousin increases.