We have investigated the mechanism of the first order transition and proton conductivity in copper rubeanate hydrates from microscopic and dynamical points of view. Three different types of neutron spectrometer-time-of-flight, backscattering, and neutron spin echo-were used to ever a wide dynamic range (1 ps to 100 ns). We found that the water molecules adsorbed in the pore are divided into "free water" having diffusion coefficients similar to those of bulk water at room temperature and "condensed water" which is about 10 times slower than bulk water owing to the interaction with the pore wall. The hydrogen atoms in the pore wall exhibited no relaxation within the measured time scales. The free water has, in the framework of the jump-diffusion model, smaller activation energy, longer residence time, and longer jump distance than bulk water. The neutron spin echo measurement revealed that the first order transition is a kind of liquid-liquid transition at which the free water is condensed on the pore surface in the low temperature phase. On cooling the condensed water, the relaxation time starts to deviate from the VFT equation around 200 K as previously observed in the water confined in nanoporous silicates. The free water plays an important role as the proton carrier but the proton conductivity is mainly governed by the number of protons provided into the adsorbed water from the pore wall.