In this letter, we address the problem of exploration and metric-semantic mapping of multi-floor GPS-denied indoor environments using Size Weight and Power (SWaP) constrained aerial robots. Most previous work in exploration assumes that robot localization is solved. However, neglecting the state uncertainty of the agent can ultimately lead to cascading errors both in the resulting map and in the state of the agent itself. Furthermore, actions that reduce localization errors may be at direct odds with the exploration task. We propose a framework that balances the efficiency of exploration with actions that reduce the state uncertainty of the agent. In particular, our algorithmic approach for active metric-semantic SLAM is built upon sparse information abstracted from raw problem data, to make it suitable for SWaP-constrained robots. Furthermore, we integrate this framework within a fully autonomous aerial robotic system that achieves autonomous exploration in cluttered, 3D environments. From extensive real-world experiments, we showed that by including Semantic Loop Closure (SLC), we can reduce the robot pose estimation errors by over 90% in translation and approximately 75% in yaw, and the uncertainties in pose estimates and semantic maps by over 70% and 65%, respectively. Although discussed in the context of indoor multi-floor exploration, our system can be used for various other applications, such as infrastructure inspection and precision agriculture where reliable GPS data may not be available.