The idea goes something like this: physics should certainly not depend on where you are at or the orientation of your measuring device. If we change this (called 'changing the frame of reference'), observable quantities should remain the same. Essentially, the point is that if I move an electron from Asia to the United States or spin it around, it remains an electron.
So, we want to encode this mathematically. Quantum mechanically, we describe systems (like this electron) with a mathematical object, the state vector (mathematical physicists, this is good enough). We need some sort of way to describe what it means to move or spin this state. Well, we can construct operators that do such things (rotation operators, translation operators, etc.); the real insight is that translation and rotation can be mapped to objects called groups. A group is a set with an operation that takes two members of the group and outputs a third (with some qualifications on the structure of the operation). Translations can be described with a group; if I drag an object three meters north and then four meters east, this is the same as dragging it five meters in a northeastern direction. Likewise, rotations also form a group.
So, say we have a state that describes an electron. When we act an operator corresponding to a rotation or translation on the state, the resulting state should also describe an electron. Mathematically, we define the description of electrons this way; they're described by the set of states that mix among themselves when acted on by operations from a specific group (in the nonrelativistic case, this is the Galilean group; in the relativistic one, this is the Poincare group).
A set of objects that transform among themselves under group operations are called a group representation. We add a couple other reasonable stipulations: there shouldn't be a subgroup of electron states that only transforms among itself; given any electron state, I should be able to move it or rotate it into any configuration I'd like. Thus, our representation is a so-called irreducible representation. Furthermore, when I rotate or translate my state, the observable predictions should remain the same (a scattering process does not care if it is done in China or Germany), which, due to the structure of quantum mechanics, imposes an additional constraint: unitarity. Thus, particles are defined as irreducible unitary representations of the Galilean/Poincare group. Particles are distinguished from one another by their quantum numbers (mass, charge, and yes, spin, among others). This is known as Wigner's classification.
Now, this imposes incredible restraints on what sort of states you can have. In relativistic and non-relativistic theory, particles have to remain particles after rotation in plain old three-dimensional space: this translates to, in technical terms, as being an irreducible unitary representation of the group SU(2), which encodes rotations in three-dimensional space (it is a subgroup of both the Galilean and Poincare groups). The "irreducible unitary" part enforces stringent qualifications on the states; you get different possible families of states, each (traditionally) labeled by half-integers: j=0,1/2,1,3/2,...
This is spin. States of non-zero j have internal degrees of freedom that mix among themselves when mathematically rotated (this is what Woit means by "in this case rotations also act on the vector values"). When you construct angular momentum from rotation (which is a fascinating discussion in its own right), this corresponds to intrinsic angular momentum.