What makes bones grow longer and bigger?

Biologically there. Is a growth plate at the end of each bone. Some bones like your femur and humerus have multiple growth plates. Your finger bones have one growth plate. A protion of this growth plate provides for length and a smaller portion provides for width. The periosteum "or the layer of tissue that surrounds the bones" also lays down bone to provide width. Bone size is largely determined by genetics.