Microwave penetration depth lambda and surface resistance at 27 GHz are measured in high quality crystals of KOs(2)O(6). Firm evidence for fully gapped superconductivity is provided from lambda(T). Below the second transition at T(p) approximately 8 K, the superfluid density shows a steplike change with a suppression of effective critical temperature T(c). Concurrently, the extracted quasiparticle scattering time shows a steep enhancement, indicating a strong coupling between the anomalous rattling motion of K ions and quasiparticles. The results imply that the rattling phonons help to enhance superconductivity, and that K sites freeze to an ordered state with long quasiparticle mean free path below T(p).