Initially a phenomenological guess to explain the periodic table. Later it was apparently proven properly with the spin-statistics theorem, physics.stackexchange.com/questions/360140/theoretical-proof-of-paulis-exclusion-principle.
And it was understood more and more that basically this is what prevents solids from collapsing into a single nucleus, not electrical repulsion: electron degeneracy pressure!
The name actually comes from "any". Amazing.
Key physical experiment: fractional quantum Hall effect.
Exotic and hard to find experimentally.
Video "The Biggest Ideas in the Universe | 17. Matter by Sean Carroll (2020)" at youtu.be/dQWn9NzvX4s?t=3707 says that no one has ever been able to come up with an intuitive reason for the proof.