The longer answer has to include what we observe and what we we have to deduce. Oersted has observed that a moving in circles electron (the current in the coil) led to a magnetic field. How that happens in detail:

Moving in circles a body experience an acceleration. Why it is important to mention this fact? Because under this acceleration the electrons spins get aligned. And the intrinsic magnetic dipole moment of the electron gets aligned together with its spin! So the magnetic field of the coil is the sum of the magnetic dipole moments of the involved electrons.