Where does particle spin come from?
Spin doesn't "come from" anywhere; it's a fundamental property that some particles have. An electron, for example has spin 1/2. Its spin is a part of its "electron-ness". The spin is inextricable from the particle's identity. If an "electron" didn't have spin 1/2, it just wouldn't be an electron at all; it would be a totally different particle. The spin is angular momentum the electron gets "just for being an electron". (Note that other particles, such as quarks, also have the same spin, though.)
Now I'll address another interpretation of the question, which is why there should be such a phenomenon as spin at all. Spin is angular momentum that some particles intrinsically possess. In order to understand why some particles can have angular momentum even when they're not moving, you need to understand what angular momentum is.
Angular momentum is the conserved quantity associated with the invariance of the physical universe under rotation. In a few words, on one hand, a rotation is not an identity transformation; in general, it changes the states it acts on; on the other hand, this change is a symmetry, and the universe's behaviour doesn't depend on how it's oriented. There is a famous theorem due to Emmy Noether that states that certain kinds of physical symmetries imply conservation laws, and it gives a formula for the conserved quantity corresponding to a symmetry. Angular momentum is the name we give to the conserved quantity corresponding to rotational symmetry.
I won't give the formula for Noether's theorem in this answer; you can find it here. But one important thing to note about the formula is that if you have a state that doesn't change under the symmetry, then the corresponding conserved quantity will be zero. If the state does change, the conserved quantity will usually be nonzero. There are certain kinds of particles we call scalar particles, which are totally unaffected by rotation. (The Higgs boson is a scalar particle.) If you have a scalar particle stationary at the origin, then a rotation does literally nothing to it. Applying the formula, we find that such a system has no angular momentum at all, and so the particle is spinless. But there are other kinds of particles that are affected by rotation, for example, electrons. Rotating an electron may change the direction of its dipole moment, for example. When we write down the state of an electron, we need to use a mathematical object called a spinor to describe its orientation, and when we perform a rotation, we change the spinor, and when we apply Noether's theorem, because the spinor changes, we get a nonzero conserved quantity, which in this case is angular momentum. So even while the electron doesn't move, it still changes, and because of that the electron has angular momentum. So the spin of the electron comes from the fact that the electron changes when it's rotated.
Now I'll address another interpretation of the question, which is why there should be such a phenomenon as spin at all. Spin is angular momentum that some particles intrinsically possess. In order to understand why some particles can have angular momentum even when they're not moving, you need to understand what angular momentum is.
Angular momentum is the conserved quantity associated with the invariance of the physical universe under rotation. In a few words, on one hand, a rotation is not an identity transformation; in general, it changes the states it acts on; on the other hand, this change is a symmetry, and the universe's behaviour doesn't depend on how it's oriented. There is a famous theorem due to Emmy Noether that states that certain kinds of physical symmetries imply conservation laws, and it gives a formula for the conserved quantity corresponding to a symmetry. Angular momentum is the name we give to the conserved quantity corresponding to rotational symmetry.
I won't give the formula for Noether's theorem in this answer; you can find it here. But one important thing to note about the formula is that if you have a state that doesn't change under the symmetry, then the corresponding conserved quantity will be zero. If the state does change, the conserved quantity will usually be nonzero. There are certain kinds of particles we call scalar particles, which are totally unaffected by rotation. (The Higgs boson is a scalar particle.) If you have a scalar particle stationary at the origin, then a rotation does literally nothing to it. Applying the formula, we find that such a system has no angular momentum at all, and so the particle is spinless. But there are other kinds of particles that are affected by rotation, for example, electrons. Rotating an electron may change the direction of its dipole moment, for example. When we write down the state of an electron, we need to use a mathematical object called a spinor to describe its orientation, and when we perform a rotation, we change the spinor, and when we apply Noether's theorem, because the spinor changes, we get a nonzero conserved quantity, which in this case is angular momentum. So even while the electron doesn't move, it still changes, and because of that the electron has angular momentum. So the spin of the electron comes from the fact that the electron changes when it's rotated.