The narrator (obviously an unreliable narrator, in the typical vein of Poe's criminally mad narrators) claims to have lived a childhood whose greatest comforts involved animals. When we learn of that in the second paragraph, we may take the statement at face value, not yet fully aware of the degree of the narrator's madness. We only know in paragraph one that he is about to die and wishes to deny his madness.
He documents his childhood as one marked by a "tenderness of heart" that caused others to ridicule him. In the animals he had as a child, he found happiness "feeding and caressing them." This would seem normal enough, but in the same paragraph, we find one telling clue: "There is something in the unselfish and self-sacrificing love of a brute, which goes directly to the heart of him who has had frequent occasion to test the paltry friendship and gossamer fidelity of mere Man." In this sentence, a reader returning to the story with full knowledge of the ending may wonder what the narrator imagines as "unselfish and self-sacrificing." How lacking in fidelity were his boyhood relations with pets: were they filled with neglect or his signature cruelty?
In the description of his childhood relationships with his pets, the narrator seeks to exonerate himself. However, as is typical in a Poe story, the attempt to convey innocence is also evidence of guilt—not necessarily in terms of actions committed but of psychological distortions that lead to later crimes.