In classical mechanics, a charge going round a circle produces a magnetic field and has a magnetic dipole moment. We can thus relate angular momentum of the charge with the magnetic dipole moment.
Moving to quantum mechanics, some particles have an intrinsic magnetic moment, the way they could be having an electric charge or a rest mass. For such particles, it helps to interpret the intrinsic magnetic moment in terms of an intrinsic angular momentum. Spin is another name to the intrinsic angular momentum.
Instead of trying to visualize an electron spinning, you may just accept the experimental fact that it has an intrinsic magnetic moment and therefore spin.
The relation between intrinsic magnetic moment and spin angular momentum is not just a mathematical construct. The spin part of angular momentum (S) is essential to ensure conservation of angular momentum. By that, I mean, orbital angular momentum (L) alone is not conserved but the sum of it (J = L + S) and spin, called total angular momentum is conserved.