Why is the static force acting on the friction force during walking? How, at every step, does it act by allowing movement?
When we walk or run we apply a pushing force against the ground. The ground applies an equal and opposite reaction force on us. See the free body diagram of a runner below. It applies as well to a walker.
The ground reaction force on the person is resolved into the static friction force parallel to the surface and the reaction force normal to the surface. During portions of walking or running the normal reaction force is greater than the gravitational force on the person in order to lift the person off the ground.
The static friction force that the ground applies to the person propels the person
forward, and is equal and opposite to the parallel force the person applies to the ground. If there were no static friction force the person will slip. Note that if the parallel component of the per pushing force on the ground exceeds the maximum possible static friction force, the person slips.
The difference between the person shown running and a person walking is the angle $θ$ is greater for the walker. When the person is standing still ($θ=90^0$) and the only forces are the persons weight and the equal and opposite normal force on the ground. There is no pushing force for friction to oppose, and so therefore no friction.
Hope this helps.