I think Piaget's biggest claim in the area of psychological and therefore educational change was his focus on children in the early childhood stages. Before Piaget, most psychologists were in the habit of studying children who were old enough to answer questions. Piaget suggested that behavioral patterns begin in the brain, and children can think before they can speak. His study focused on infants and very young children (namely, his own at first).
Next, Piaget began to shift away from a focus on environmental factors as the motivators for behavior. He identified essentially four stages of cognitive development that he suggested every human goes through, and provided approximate time frames for each. Through these stages of cognitive development teachers of early childhood (and parents of infants) were able to approach learning, discipline, and communication in a completely different way.
Mainly, from birth until about age 7, children are only in the sensory-motor and pre-operational stages. This means in pre-school, kindergarten and 1st grades, most students are unable to think abstractly. Every thought is driven by what they can see and touch. Cause and effect are very difficult concepts. Memorization of facts and skills does not connect with understanding.
Basically, Piaget can be credited with the fairly recent movement in early childhood education from a teacher-centered approach to a student-centered approach. Classrooms today are much more interactive, full of hand-held manipulatives, and much of the education happens through self-discovery and play time. Because at the early stages, children are inherently ego-centric, curriculums were developed to move from the child, outward. Learning is built on previous knowledge that actually grows with the child's cognitive ability to understand.