How Children Learn Language is a book that covers this topic. This is the book's summary:
Demonstrating how children learn to produce and distinguish between sounds, and their acquisition of words and meanings, this book explains their incredible mastery of language. William O'Grady provides readers with an overview not only of the language acquisition process itself, but also of the ingenious experiments and techniques that researchers use to investigate this mysterious phenomenon.
How Babies Talk: The Magic and Mystery of Language in the First Three Years of Life is a related book. The summary is:
In their first three years of life, babies face the most complex learning endeavor they will ever undertake as human beings: They learn to talk. Now, as researchers make new forays into the mystery of the development of the human brain, Golinkoff and Hirsh-Pasek, both developmental psychologists and language experts, offer parents a powerfully insightful guidebook to how infants—even while in the womb—begin to learn language. Along the way, the authors provide parents with the latest scientific findings, developmental milestones, and important advice on how to create the most effective learning environments for their children. This book takes readers on a fascinating, vitally important exploration of the dance between nature and nurture, and explains how parents can help their children learn more successfully.
Finally, Child Language: Acquisition and Growth is a Cambridge linguistics textbook. This is its focus:
Exploring language development from birth, this accessible textbook introduces the field of child language acquisition, establishing key theoretical debates. It considers what characteristics of the human mind make it possible to acquire language and to what extent acquisition is biologically programmed and influenced by our environment. It inquires as to what makes second language learning (in adulthood) different from first language acquisition; and whether the specific stages in language development are universal across languages. It will be a key text for courses in linguistics, developmental psychology and cognitive science.