As for why the spin-statistics theorem is true, the answer is that, in a sense, we do not really know. This is because, although we have rigorous mathematical proofs that it is true, they rely on arguments that are very technical in nature, so they provide no real intuitive insight into why the theorem is true. (This theorem is actually really notorious for this; people have been trying for a long time to improve on the situation, but have yet to succeed in coming up with a satisfactory elementary proof of it.)
Which, in turn, is a consequence of the spin-statistics theorem.
As for why the spin-statistics theorem is true, the answer is that, in a sense, we do not really know. This is because, although we have rigorous mathematical proofs that it is true, they rely on arguments that are very technical in nature, so they provide no real intuitive insight into why the theorem is true. (This theorem is actually really notorious for this; people have been trying for a long time to improve on the situation, but have yet to succeed in coming up with a satisfactory elementary proof of it.)