Imagine you are at the center of a carrousel holding a magneto meter. You represent the nuclei of an atom. Now on one of the horses a magnetic bar is fixed. You can measure the influence of the magnetic bar on your position with the magneto meter. This is the spin dipole contribution to your measurement.
If we now add a charged ball to another horse, a second magnetic field will be generated due to the charge going round in a circle because the carrousel is spinning. This is the orbital contribution.
If now a child enters the carrousel and takes the ball and starts moving around with it. The changes is real the ball will reach you. This is the Fermi term to the hyperfine field.