We present the results of C-13 NMR experiments in an organic superconductor with localized Fe spins beta ''-(BEDT-TTF)(4)[(H3O)Fe(C2O4)(3)] center dot C6H5Br. We reveal the antiferromagnetic coupling between Fe d spins and pi spins, which creates an exchange field antiparallel to the external field direction at the pi electrons. In addition to the static effects of Fe spins, we show from the nuclear spin-lattice relaxation rate measurement that the magnetic fluctuations generated by Fe spins are suppressed at low temperatures and high magnetic fields. These conditions are suitable to stabilize the field-induced superconductivity by the field compensation mechanism. After the suppression of Fe-spin dynamics by a magnetic field of 19 T, we observed the underlying pi-electron contribution. We discuss a possible anomaly in the pi-electron system.