>>13723232it was proven mathematically, that's part of what Roger Penrose won the nobel for
a simpler intuitive explanation goes like this: what holds the neutron star from collapsing normally is the outward pressure from nuclear interactions. in a neutron star this is the exchange of pions, but there is also a phase where neutrons become so dense/hot that they change into a "quark star" where the neutrons disassociate (into a "quark gluon plasma") and the pressure holding them from collapsing is mediated by gluons
now, whether it is pions or gluons, the argument is the same. imagine we're in some distant reference frame far away from the neutron/quark star, and just assume it's approximately spherically symmetric. in GR when a black hole forms, basically some small region of space exceeds the threshold for the formation and a horizon appears, and it grows out from there (by eating stuff). so when a neutron/quark star starts turning into a black hole, a little horizon forms inside it. now, because there is a horizon there, that means nothing can be shot from the interior to the exterior of the horizon. so the pions or gluons that used to be propagating from the middle of the neutron/quark star outward to supply the outward pressure can't actually make it out of the horizon to supply any pressure. so the exterior loses the pressure that was supporting it from collapsing, and thus the quarks/neutrons just fall in at free fall.
now, inside the black hole the curvature is even HIGHER than in the interior. this means the stuff even inside the black hole can't manage to shoot any gluons out to supply pressure, and it gets more and more severe the closer you get to the middle. so everything inside the black hole is basically in free fall toward the center of where the initial horizon formed. that's the singularity -- the place toward where everything is in free fall since its gravitational pull makes it impossible for any particle to exert outward pressure
