Exploration of unknown environments is a fundamental problem in robotics and an essential component in numerous applications of autonomous systems. A major challenge in exploring unknown environments is that the robot has to plan with the limited information available at each time step. While most current approaches rely on heuristics and assumption to plan paths based on these partial observations, we instead propose a novel way to integrate deep learning into exploration by leveraging 3D scene completion for informed, safe, and interpretable exploration mapping and planning. Our approach, SC-Explorer, combines scene completion using a novel incremental fusion mechanism and a newly proposed hierarchical multi-layer mapping approach, to guarantee safety and efficiency of the robot. We further present an informative path planning method, leveraging the capabilities of our mapping approach and a novel scene-completion-aware information gain. While our method is generally applicable, we evaluate it in the use case of a Micro Aerial Vehicle (MAV). We thoroughly study each component in high-fidelity simulation experiments using only mobile hardware, and show that our method can speed up coverage of an environment by 73% compared to the baselines with only minimal reduction in map accuracy. Even if scene completions are not included in the final map, we show that they can be used to guide the robot to choose more informative paths, speeding up the measurement of the scene with the robot's sensors by 35%. We validate our system on a fully autonomous MAV, showing rapid and reliable scene coverage even in a complex cluttered environment. We make our methods available as open-source.