So, we apply a point load to the end of the cantilever beam and can see that for external equilibrium, there must be shear reaction at the support. (Leaving aside for now the moment reaction that also exists.) We have an intuitive understanding of this Newtonian action-reaction from our daily lives.
So, the next question is, how is the load 'getting' from the point of application to the support? We can assess that by looking at internal equilibrium. If we start from the right end of the beam (where the load is applied) and cut the beam at any section, there must be a shear reaction for the beam segment to be in equilibrium. We can see this using the same logic as for the external equilibrium.
This illustrates that the applied load creates a shear force at every point along the beam.
The example you gave from the human body is perhaps a difficult one for developing this intuition, given the complexity of muscle, tendon, and bone. However, you mention that if you you press down on your hand you don't feel the "pressure" anywhere else, so this may be about distinguishing between internal and external equilibrium. Pressure is external. The beam (or your hand) only feels that external pressure at the point of load application and at the support. However, if you hold out your forearm and hand rigidly and push down on the tips of your fingers, you will feel muscles and tendons reacting to the load to maintain equilibrium.