How quantum spin causes magnetism?

Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum.

Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics. These magnetic moments can be experimentally observed in several ways, e.g. by the deflection of particles by inhomogeneous magnetic fields in a Stern–Gerlach experiment, or by measuring the magnetic fields generated by the particles themselves.

The intrinsic magnetic moment $mu$ of a spin $1/2$ particle with charge $q$, mass $m$, and spin angular momentum $mathbf{S}$, is $$mu=frac{g_sq}{2m}mathbf{S}$$ where the dimensionless quantity $g_s$ is called the spin g-factor. For exclusively orbital rotations it would be $1$ (assuming that the mass and the charge occupy spheres of equal radius).