Multistep prediction models are essential for the simulation and model-predictive control of dynamical systems. Verifying the safety of such models is a multi-faceted problem requiring both system-theoretic guarantees as well as establishing trust with human users. In this work, we propose a novel approach, ReLiNet (Recurrent Linear Parameter Varying Network), to ensure safety for multistep prediction of dynamical systems. Our approach simplifies a recurrent neural network to a switched linear system that is constrained to guarantee exponential stability, which acts as a surrogate for safety from a system-theoretic perspective. Furthermore, ReLiNet's computation can be reduced to a single linear model for each time step, resulting in predictions that are explainable by definition, thereby establishing trust from a human-centric perspective. Our quantitative experiments show that ReLiNet achieves prediction accuracy comparable to that of state-of-the-art recurrent neural networks, while achieving more faithful and robust explanations compared to the model-agnostic explanation method of LIME.