The phrase "board book" refers to a type of binding used to make a book. The pages and the cover are all from cardboard. The whole book is durable instead of just the cover, making it ideal for young children.
A story board book is a very short story with pictures, bound as a board book. Board books are sometimes written to introduce a child to concepts, like colors or animals. Story board books tell the child a tale or a poem, and each cardboard page has an accompanying picture.
The Walker section of "Walker story board book" refers to a publisher, Walker Books. Their children and young adult division is known as Walker Books in the United Kingdom and Australia and as Bloomsbury and Walker Books for Young Readers with the sister company Candlewick Press in the United States.