Large-scale measurements linking genetic background to biological function have driven a need for models that can incorporate these data for reliable predictions and insight into the underlying biochemical system. Recent modeling efforts, however, prioritize predictive accuracy at the expense of model interpretability. Here, we present LANTERN (https://github.com/usnistgov/lantern), a hierarchical Bayesian model that distills genotype-phenotype landscape (GPL) measurements into a low-dimensional feature-space that represents the fundamental biological mechanisms of the system while also enabling straightforward, explainable predictions. Across a benchmark of large-scale datasets, LANTERN equals or outperforms all alternative approaches, including deep neural networks. LANTERN furthermore extracts useful insights into the landscape including its inherent dimensionality, a latent space of additive mutational effects, and novel metrics of landscape structure. LANTERN facilitates straightforward discovery of fundamental mechanisms in GPLs, while also reliably extrapolating to unexplored regions of genotypic-space.