spin-statistics connection
Two facts about particles seem totally unrelated at first glance. One is about rotation: a particle's spin is either a half-integer or a whole integer. The other is about company: some particles refuse to share a state (the antisocial fermions) while others love to pile in together (the sociable bosons). Astonishingly, these two facts are locked together. Half-integer spin always goes with the antisocial behavior, and whole-integer spin always goes with the sociable behavior. The rule that ties spin to social behavior is the spin-statistics connection.
What does swapping have to do with it? Quantum mechanics treats identical particles as truly interchangeable, so when you imagine exchanging two of them their combined wavefunction can at most pick up a sign. For bosons the wavefunction stays the same under a swap (a plus sign), which lets two of them happily occupy the same state. For fermions it flips sign (a minus sign), and a state that equals minus itself must be zero — meaning two identical fermions in the same state is simply forbidden. That sign flip is the Pauli exclusion principle in disguise. The deep result, proved by relativistic quantum field theory, is that the swap sign is forced to match the spin: half-integer spin gives the minus, integer spin gives the plus.
This connection is one of the most important theorems in physics, because so much rests on it. It is why matter is rigid and atoms have shells (fermion electrons must keep apart), and why lasers and force fields can be intense (bosons can stack up). It cannot be derived from non-relativistic quantum mechanics alone; it genuinely requires combining quantum theory with special relativity, which is one reason the result is considered so profound. In short, a particle's most abstract rotational property dictates one of the most concrete facts about how the world is built.
Swap two electrons in an atom and their joint wavefunction flips sign; demand both be in the same state and that wavefunction must equal its own negative, hence zero — so the state is forbidden, and electron shells are born.
The sign you pick up when swapping two particles is fixed by their spin — that is the whole theorem.
The connection is a theorem of relativistic quantum field theory, not an arbitrary rule; it cannot be obtained from ordinary non-relativistic quantum mechanics on its own.