It seems to be related to exchange interaction, but I can't penetrate the Wikipedia article. What has the Pauli exclusion principle to do with indistinguishability?
Indistinguishableness of particles is formulated in QM in terms of the total wave function symmetry. Then the wave functions can be symmetric or antisymmetric on their arguments. In case of antisymmetry ($\psi(x_1,x_2) = -\psi(x_2,x_1))$ the particles are called fermions and $\psi(x_1,x_2=x_1) = 0$. They say "the particles cannot occupy the same state". It's a feature of nature, an experimental fact.
The exchange interaction is, on the contrary, one of the approximate low-energy consequences of the Pauli exclusion principle and the identical character of the particles.
The fundamental justification of these facts is offered by quantum field theory. Relativity requires observables to be linked to regions of spacetime - so that they don't communicate over spacelike separations.
It follows that the observables must be - or at least, in the simplest cases such as the Standard Model and its subsets, they are - functionals of quantum fields. Spacelike separated fields either commute or anticommute with each other - these are the only two ways how to make sure that they don't interact faster than light.
The Pauli exclusion principle only applies to half-integral-spin particles because of Pauli's spin-statistics theorem,
The page above also has a convincing bogus proof and a proof.