This is answered in detail in Weinberg's QFT book, vol. I, Chapter 5.
Briefly, to satisfy Lorentz invariance of the S-matrix, one needs (5.1.3), hence (by the form of $H$) that the spacelike (anti)commutators of the basic fields vanish. To construct the fields from the physical Poincare irreps, one needs to take appropriate linear combination of the creation and annihilation fields to ensure that the spacelike (anti)commutators vanish (5.1.32), and this is found to require the commutator for integral spin and the anticommutator for half-integral spin (5.7.28).
For full details see the discussion around (5.1.29-33) and (5.7.28-29).