Well stability relate to the bodily bounded by the abdominal wall, pelvis, lower back, diaphragm, and its ability to stabilize the body during movement.
The main muscle involves include the transversus abdominal, internal and external oblique, quadratus lumborum, and the diaphragm.
Diaphragm is the main muscle for breathing in the human, and breathing is so important in providing core for moving and lifting. It is the action of this muscles contracting together upon the incompressible content of the abdominal activity.
Once you start improving core stability, the benefit will translate into both sports performances and daily life.