Best mathematical explanation: Section "Spin comes naturally when adding relativity to quantum mechanics".

Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has an amazing summary of the preceding chapters the spin value has a relation to the representations of the Lorentz group, which encodes the spacetime symmetry that each particle observes. These symmetries can be characterized by small integer numbers:

As usual, we don't know why there aren't elementary particles with other spins, as we could construct them.