The spin-statistics theorem relates the bosonic/fermionic nature of a particle to the type of exchange symmetry of indistinguishable many-particle wavefunctions. Possibly it is easiest to consider the consequences of this exchange symmetry in 2-particle systems.

For fermions, states must be antisymmetric so that $$ \vert \Psi_-\rangle = \frac{1}{\sqrt{2}}\left(\vert \psi\rangle_1\vert\phi\rangle_2 - \vert \psi\rangle_2\vert\phi\rangle_1\right)\, . $$ In particular, if $\vert\phi\rangle=\vert\psi\rangle$, $\vert\Psi\rangle=0$, thus enforcing the Pauli condition that two fermions cannot occupy the same state.

Contrariwise, for bosons $$ \vert\Psi_+\rangle=\frac{1}{\sqrt{2}}\left(\vert \psi\rangle_1\vert\phi\rangle_2 +\vert \psi\rangle_2\vert\phi\rangle_1\right)\, $$ permutation symmetry of identical bosonic states leads to a bunching effect demonstrated experimentally by the Hanbury-Brown-Twiss effect.

Thus, permutation symmetry and the related eigenvalue of the exchange operator is compatible with the observed experimental behaviour of fermions and bosons under exchange.

The spin-statistics theorem relates the bosonic/fermionic nature of a particle to the type of exchange symmetry of indistinguishable many-particle wavefunctions. Possibly it is easiest to consider the consequences of this exchange symmetry in 2-particle systems.

For fermions, states must be antisymmetric so that $$ \vert \Psi_-\rangle = \frac{1}{\sqrt{2}}\left(\vert \psi\rangle_1\vert\phi\rangle_2 - \vert \psi\rangle_2\vert\phi\rangle_1\right)\, . $$ In particular, if $\vert\phi\rangle=\vert\psi\rangle$, $\vert\Psi\rangle=0$, thus enforcing the Pauli condition that two fermions cannot occupy the same state.

Contrariwise, for bosons $$ \vert\Psi_+\rangle=\frac{1}{\sqrt{2}}\left(\vert \psi\rangle_1\vert\phi\rangle_2 +\vert \psi\rangle_2\vert\phi\rangle_1\right)\, $$ permutation symmetry of identical bosonic states leads to a bunching effect demonstrated experimentally in Hanbury-Brown-Twiss-type of experiments.

Thus, permutation symmetry and the related eigenvalue of the exchange operator is compatible with the observed experimental behaviour of fermions and bosons under exchange.