We present the results of C13 NMR experiments in an organic superconductor with localized Fe spins β″-(BEDT-TTF)4[(H3O)Fe(C2O4)3]·C6H5Br. We reveal the antiferromagnetic coupling between Fe d spins and π spins, which creates an exchange field antiparallel to the external field direction at the π 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 π-electron contribution. We discuss a possible anomaly in the π-electron system.