Baby tuckoo is Stephen Dedalus.
When Stephen is little, his father tells him a story about a nice little boy who meets a cow. He says it in the kind of language that many parents would use when talking to a child—a kind of affectionate babytalk. This is how the book opens, with a moocow coming down the road and meeting baby tuckoo.
Then the viewpoint switches, and Stephen remembers seeing his father as he told him the story. He remembers the beard on his face. He remembers the words his father spoke. He says that he, himself, was baby tuckoo.
This introduction takes the reader to the earliest memories that Stephen has. It helps to show the characteristics of his father before showing the rest of his family. It's clear that they're close and warm; his father and the stories he told were important to Stephen in his formative years.