Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has a good 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.