The "person-centered approach" began in the 1940s with psychologist Carl Rogers. He is considered a pioneer in the field of modern psychology, especially because of his alignment with the psychological movement called humanism.
Humanism came about as a reaction to behaviorism, the earlier and still popular subfield of psychology in which it's believed that our behavior is determined by how we've been conditioned throughout life, based on what we've experienced in our environments. (Note: That's a very simplified explanation! For more info, feel free to check out behaviorism as a key word).
While there are a lot of true and powerful aspects of behaviorism, many academics, scholars, professionals, and philosophers have argued that it also devalues human life and choice-making. It has been distilled to the idea that people don't have free will. While behaviorism is frequently applied to helping animals with behavior, it has not quite encompassed the entirety of human experience.
This is where humanism was born. Rogers and others like him (like Maslow!) believed in the idea that all people deserved dignity and respect. When we are empowered to make the right decisions, it becomes easier and more possible for someone to choose the appropriate actions for a circumstance.
Notice how I didn't say it's about choosing the "perfect" actions?
We do the best we can with what we've got. This is why a person-centered approach is important in any business management as well. If an employee is treated like a tool, they resent their leadership; they don't just "work to work" because they are invested in their own lives and needs as well (and rightfully so - that's how a person survives both physically and psychologically). When their needs are met and they are invited to contribute in a manner which respects their dignity and basic human needs, we see enhanced contributions.
Of course, I often word lingo so that leaders who are reading this will see the benefit of treating employees well. However, there's also more to life than status and money; it feels good to know that you've also contributed to the lives of others. People deserve decent treatment because in the end, we're all equal.