Multiclass machine learning classification has many potential applications for both clinical neuroscience and data-driven biomarker discovery. However, to be applicable in these contexts, the machine learning methods must provide a degree of insight into their decision-making processes during training and deployment phases. We propose the use of a hierarchical architecture with layer-wise relevance propagation (LRP) for explainable multiclass classification of neural states. This approach provides both local and global explainability and is suitable for identifying neurophysiological biomarkers, for assessing models based on established domain knowledge during development, and for validation during deployment. We develop a hierarchical classifier composed of multilayer perceptrons (MLP) for sleep stage classification using rodent electroencephalogram (EEG) data and compare this implementation to a standard multiclass MLP classifier with LRP. The hierarchical classifier obtained explainability results that better aligned with domain knowledge than the standard multiclass classifier. It identified α (10-12 Hz), 0 (5-9 Hz), and β (13-30 Hz) and 0 as key features for discriminating awake versus sleep and rapid eye movement (REM) versus non-REM (NREM), respectively. The standard multiclass MLP did not identify any key frequency bands for the NREM and REM classes, but did identify δ (1-4 Hz), 0, and α as more important than β, slow-y (31-55 Hz), and fast-y (65-100Hz) oscillations. The two methods obtained comparable classification performance. These results suggest that LRP with hierarchical classifiers is a promising approach to identifying biomarkers that differentiate multiple neurophysiological states.