The Born interpretation of the wavefunction leads to boundary conditions in each spatial degree of freedom, which then leads to quantum numbers when solving the Schrodinger equation. That is only certain solutions of the S.E. obey the boundary conditions, and these solutions have a quantum number that falls out of the math. For example, a 1D particle in a box (no spin) has 1 quantum number, n, such that the wavefunction is zero at the walls: psi(x) = 1/sqrt(2Pi) * sin ( n Pi x / L).

For the spin, we do not know what the spin operator or spin coordinate looks like, but we know it behaves like orbital angular momentum, giving another quantum number.

So for each quantum particle, there are 3 spatial degrees of freedom (which we can describe mathematically) and 1 spin coordinate (which we cannot describe mathematically but is there), so 4 degrees of freedom for each quantum particle.