Quantum mechanics postulates that the state space of a system has the structure of a Hilbert space. To investigate the statistical properties of a collection of N particles, we can take the state space of each individual particle and take their tensor products to get the collection's state space. This is called a Fock space.
However, experimentally, we find that the Fock space of a system composed of N identical particles is actually smaller than this full tensor product. Specifically, the particles must be "indistinguishable"; this is formalized using permutation operators, which are defined as the natural action of re-ordering the tensor product. A composite system's states are then restricted to the intersection of the +1/-1-eigenspaces of all the permutation operators (more on the +1/-1 thing later).
For example, a 2-particle system where the single-particle state space is spanned by a basis {a, b} will have a tensored state space spanned by {aa, ab, ba, bb}. The permutation operator on this space exchanges particles 1 and 2, meaning Paa := aa, Pab := ba, Pba := ab, and Pbb = bb. The indistinguishability criterion is then that for any state x, Px = +/-x. For 2 particles, this is satisfied by aa, bb, and ab+ba for +1, and ab-ba only for -1.
Now, if the criterion is indistinguishability, a natural question would be why we don't just take the +1 eigenspace. This is because the Hilbert space is actually too large; states that only differ in (complex) norm represent the same physical space. Though we work in the Hilbert space for the conveniences of linearity, the actual physical state space requires it to be projectively reduced. (Actually, it's even more complicated because of density matrices, but I'll skip over that.) Reducing to the -1 eigenspace also produces a self-consistent theory of indistinguishable particles, and it so happens to also correctly describe fermions, where the +1 eigenspace describes bosons.
The physical reason why this indistinguishability criterion applies is because constructing a multi-particle state from the single-particle states is actually an artifice. There really are no particles; they are just excitations of a common underlying quantum field, and that it the cause of these "quantum correlations". Particles do drop out of the QFT formalism, but only in certain limiting cases, but that's why you do end up with experimentally verifiable theories from the Fock approach.
I never studied anyons in detail, so the following is just my high-level understanding. In the Fock approach, the only eigenvalues allowed for a permutation operator are +1 and -1. But by going down to the level of the quantum field, you can construct anyonic theories, where the equivalent of the permutation operator can be any arbitrary phase.
What's the physical relevance of these? I see them as explorations of some of those projective aspects of quantum mechanics, in similar vein to the Aharonov-Bohm effect.