In terms of representation theory, what has happened is that the two conjugate 2-dimensional spin representations of the spin group SU(2) = Spin(3) (as it sits inside the 3-dimensional Clifford algebra) have tensored to produce a 4-dimensional representation. The 4-dimensional representation descends to the usual orthogonal group SO(3) and so its objects are tensors, corresponding to the integrality of their spin.  The 4-dimensional representation decomposes into the sum of a one-dimensional trivial representation (singlet, a scalar, spin zero) and a three-dimensional representation (triplet, spin 1) that is nothing more than the standard representation of SO(3) on . Thus the "three" in triplet can be identified with the three rotation axes of physical space.