An explainable intelligent framework for cyber anomaly mitigation of cyber-physical inverter-based systems is presented. Smart inverter-based microgrids essentially constitute an extensive communication layer that makes them vulnerable to cyber anomalies. The distributed cooperative controllers implemented at the secondary control level of such systems exchange information among physical nodes using the cyber layer to meet the control objectives. The cyber anomalies targeting the communication network may distort the normal operation therefore, an effective cyber anomaly mitigation technique using an artificial neural network (ANN) is proposed in this paper. The intelligent anomaly mitigation control is modeled using adynamic recurrent neural network that employs a nonlinear autoregressive network with exogenous inputs. The effects of false data injection to the distributed cooperative controller at the secondary control level are considered. The training data for designing the neural network are generated by multiple simulations of the designed microgrid under various operating conditions using MATLAB/Simulink. The neural network is trained offline and tested online in the simulated microgrid. The proposed technique is applied as secondary voltage and frequency control of distributed cooperative control-based microgrid to regulate the voltage under various operating conditions. The performance of the proposed control technique is verified by injecting various types of false data injection-based cyber anomalies. The proposed ANN-based secondary controller maintained the normal operation of microgrid in the presence of cyber anomalies as demonstrated by real-time simulations on a real-time digital simulator OPAL-RT.