We report high-resolution neutron scattering measurements of the low energy spin fluctuations of KFe_{2}As_{2}, the end member of the hole-doped Ba_{1-x}K_{x}Fe_{2}As_{2} family with only hole pockets, above and below its superconducting transition temperature T_{c} (∼3.5 K). Our data reveal clear spin fluctuations at the incommensurate wave vector (0.5±δ, 0, L), (δ=0.2) (1-Fe unit cell), which exhibit L-modulation peaking at L=0.5. Upon cooling to the superconducting state, the incommensurate spin fluctuations gradually open a spin gap and form a sharp spin resonance mode. The incommensurability (2δ=0.4) of the resonance mode (∼1.2 meV) is considerably larger than the previously reported value (2δ≈0.32) at higher energies (≥∼6 meV). The determination of the momentum structure of spin fluctuation in the low energy limit allows a direct comparison with the realistic Fermi surface and superconducting gap structure. Our results point to an s-wave pairing with a reversed sign between the hole pockets near the zone center in KFe_{2}As_{2}.