Question

Why does a swing's speed increase on pushing the ground harder with feet ?

Answer 1

When the swing will be at the mean position then there will be kinetic energy as the energies related to the pendulum mechanism are kinetic and potential energy. These energies keep on changing along with the motion of the swing. Kinetic energy will be zero at the top and maximum at the bottom. On the other hand, the potential energy will be minimum at the bottom of the swing and maximum at the top. Force is applied to push the ground harder with the feet which increase the speed of the swing. When someone pushes you, your kinetic energy increases and when you push the ground with your feet then your potential energy increases which basically is the reason why the swing’s speed increases. 

Answer 2

There are two energies associated with the swing motion that are kinetic and potential energies. They keep interchanging when the swing moves. Kinetic energy is at mean positions while potential energy is at extreme positions of the swing. Kinetic energy is associated with velocity or speed of the swing and potential energy depends upon the height of the swing.
When someone pushes the ground harder with feet the potential energy increases which on changing with kinetic energy during motion ultimately increase the speed of the swing.