When formulating a physical theory, one usually begins with a set of axioms. The theory itself will be just as useful as its axioms are accurate. In particular, when dealing with a supposedly fundamental theory, the following set of axioms seems natural and well-motivated by experiments:
There is a set of fundamental entities, referred to as particles, which cannot be further subdivided. Any set of particles can be combined to form more complex systems.
This "combining" of particles is commutative (an electron plus a positron is the same as a positron plus an electron), and there is an inverse-operations (i.e., to each particle we can associate an anti-particle).
A few other technical but natural assumptions regarding how systems of particles compose.
One can prove (Deligne's theorem on tensor categories, cf. Ref 1) that the most general mathematical structure that describes a physical model with these properties is a super-Lie algebra. More precisely, particles can be thought of as irreducible representations of such an algebra, in the sense of Wigner, and where "composing particles" is just a tensor product. Needless to say, it may very well be the case that the odd part of the algebra is trivial, and indeed no experiment has ever measured a superpartner. (The theorem does not imply that supersymmetry must exist; rather, it says that supersymmetry is the only possible extension of conventional QFT if we want to keep the concept of particle, as opposed to, say, strings).
If one further assume that it makes sense to evolve these particles in time so as to talk about scattering (and that the scattering matrix satisfies some natural assumptions, such being non-trivial), then it follows from a theorem due to Haag, Łopuszański, and Sohnius (Ref. 2), that the super-Lie algebra is a direct sum of the super-Poincaré algebra and a reductive (bosonic) Lie algebra, the algebra of internal symmetries (colour, isospin, etc.). As before, particles are to be thought of as irreducible representations of the symmetry algebra. The details can be found in this PSE post (the odd part of the algebra, if non-trivial, organises particles into supermultiplets).
As described therein, particles can be further organised according to the effect of the fundamental group of the rotation group, to wit, $\mathbb Z_2$. If a particle transforms trivially, it is said to be a boson, and if non-trivially, a fermion. In other words, bosons are invariant under a $2\pi$ rotations, while fermions acquire a negative sign. This is not the content of the spin-statistic theorem, but rather a tautology that follows from the definition of boson/fermion.
Once classified the most general particle content of a theory, the next step is to construct specific models for them. The most general technique we have is known as quantum field theory, where one embeds these particles into fields, that is, objects with simple transformations under the (super-)Poincaré group. This allows us to construct models in a straightforward manner (Lagrangian field theory). Without fields, it is very complicated to construct interactions that satisfy the required assumptions (unitarity, covariance, etc.).
Let me stress the following: particles are described by representations of (the universal conver of) $\mathrm{SO}(d-1)$, while fields are described by representations of (the universal cover of) $\mathrm{SO}(1,d-1)$. In both cases it makes sense to speak of bosons/fermions, but the spin-statistics theorem concerns the latter, not the former. What's more, the spin-statistics theorem does not say that bosonic fields must commute, and fermionic fields anti-commute. Rather, it says the following: if bosonic fields anti-commute for spacelike separations, then the field must be trivial (and, similarly, so must be fermionic fields if they commute). In other words, we do not say that (fermions) bosons must (anti-) commute; instead, we say that the other possibility is forbidden. If we further assume that these are the only two possibilities, then we do get the quoted result: bosons must commute and fermions anti-commute.
It is important to note that other options besides $ab\pm ba$ have been considered in the literature, and while they make sense from an algebraic point of view, they do not tend to work well from a physics point of view. I do like to think that the fundamental reason these two possibilities are the only ones allowed by nature is the aforementioned Deligne theorem. That being said, the canonical proof of the spin-statistics theorem as quoted above can be found in Ref.3 (which addresses the $d=4$ case only, but the argument admits a natural extension to arbitrary $d$), and in Ref.4 (this time for arbitrary $d$, but using a more involved argument). In short, the higher-dimensional spin-statistics theorem is identical to the $d=4$ case: bosonic fields cannot anti-commute, and fermionic fields cannot commute.
References.
Supersymmetry and Deligne's Theorem, Urs Schreiber, https://www.physicsforums.com/insights/supersymmetry-delignes-theorem/
Weinberg's QFT, Vol.III.
Wightman's PCT, spin-statistics, and all that.
Causality, antiparticles, and the spin-statistics connection in higher dimensions, S. Weinberg, https://doi.org/10.1016/0370-2693(84)90812-8 and Causal fields and spin-statistics connection for massless particles in higher dimensions, N. Ohta, https://doi.org/10.1103/PhysRevD.31.442.
