As mentioned in @kalle's comment, the Pauli exclusion principle is a consequence of the general principles of relativistic quantum field theory (QFT). It is part of the spin-statistics theorem, which says:
particles with half-integer spin (like protons, neutrons, and electrons) must be fermions (that is, must obey the Pauli exclusion principle)
particles of integer spin (like photons) must be bosons.
"Spin" refers to the particle's intrinsic angular momentum expressed in units of Planck's constant $\hbar$. Protons, neutrons, and electrons have spin $1/2$, which means they have intrinsic angular momentum equal to $\hbar/2$.
Actually, I oversimplified a bit. QFT is formulated in terms of fields, not particles. Particles are phenomena that QFT predicts. What the spin-statistics theorem actually says is that half-integer spin fields must be fermion fields, and integer spin fields must be boson fields. Particles are manifestations of fields (with lots of caveats), but for the sake of keeping this answer brief, I'll stick with the original simplification and express things in terms of particles instead of fields.
Saying that a particle is a fermion means by definition that the wavefunction of $N$ of those particles is antisymmetric with respect to their permutations. This is not a theorem or something to be explained; it is simply a definition. The theorem, the thing to be explained, is that spin-$1/2$ particles (like electrons) are necessarily fermions — which is another way of saying that spin-$1/2$ particles necessarily obey the Pauli exclusion principle.
(Similarly, saying that a particle is a boson means that the wavefunction of $N$ of those particles is symmetric with respect to their permutations. Again, this is just a definition; the thing explained by the theorem is that integer-spin particles, like photons, must be bosons.)
The spin-statistics theorem is a consequence of a few general principle of relativistic QFT. One of the key principles is that the total energy of the system cannot be less than some finite lower bound (conventionally taken to be zero). The lowest-energy state is, by definition, the vacuum state representing empty space. Even in the simplest possible model that only includes a single relativistic spin-$1/2$ field, imposing this seemingly-innocuous principle leads to the conclusion that the field must be a fermion — in other words, that the associated particles must obey the Pauli exclusion principle.
The key difference between half-integer spin and integer spin is that the kinetic term in the Hamiltonian (energy operator) for a field with half-integer spin has an odd number of derivatives, while the kinetic term for a field with integer spin has an even number of derivatives. This is related to why the wavefunction must be antisymmetric in the first case and symmetric in the second case.
Importantly, the spin-statistics theorem relies on relativistic QFT, which means it relies on Lorentz symmetry, the symmetry of flat spacetime. (Generalizing the spin-statistics theorem to curved spacetime is a more recent effort; it's even trickier.) There is no such theorem in non-relativistic quantum mechanics. In non-relativistic quantum mechanics, we simply impose by hand that spin-$1/2$ particles must be fermions, because we know that real-world applications of non-relativistic quantum mechanics are really just approximations to relativistic QFT. This is important because it implies that we cannot understand the reason for the Pauli exclusion principle using intuition based on non-relativistic quantum physics. Relativistic QFT is essential.
The question requests intuition. Transcribing a theorem into intuition is often difficult, especially when the theorem relies on the framework of relativistic QFT. In the case of the spin-statistics theorem, I don't have an intuitive explanation that I would consider to be satisfactory. What I've tried to do in this answer is simply convey two related points:
There is a deeper reason for the Pauli exclusion principle. Given the general principles of relativistic QFT, it can be proven. The classic book is PCT, Spin and Statistics, and All That by Streater and Wightman. (It's very mathematical.) The same general principles of relativistic QFT also imply CPT symmetry, which implies the existence of antiparticles. The same book proves this theorem, too. (They called it PCT. Most of us today call it CPT. Same thing.) There is a more recent proof whose assumptions are even more natural (but the proof is even more technical).
The deeper reason relies on relativistic QFT. It cannot be understood using only non-relativistic quantum mechanics, where we merely enforce it by hand.
