The procedure which introduces spin formally, requires you to know some group theory : irreducible representations (IR) of a symmetry group etc. Do you know this ? If not, answering your question is going to be difficult.
In short : spin is assigned because of symmetry. More specifically, in QM the following concepts are invariant under spatial rotations:
1) the normalization of the wavefunction
2) the expectation value of any observable
3) the Hamiltonian
For this symmetry to be respected, the wavefunctions need to "behave" in a certain way under such rotations. Mathematically one says : the wavefunctions need to transform under certain IR of the rotational symmetry group.
One set of wavefunctions that respects this demand are the spinors. They are defined based upon their behaviour under rotatios. If A is a spinor and you rotate it 360 degress you get -A.
To read the full story, go check https://www.physicsforums.com/blogs/marlon-13790/what-is-spin-152/
marlon
