North Atlantic sea surface temperatures (NASST), particularly in the subpolar region, are among the most predictable locations in the world's oceans. However, the relative importance of atmospheric and oceanic dynamics on their variability at decadal and longer timescales remain uncertain. Neural networks (NNs) are trained to predict extreme warm and cold NASST states in the Community Earth System Model 1 (CESM1) using oceanic and atmospheric predictors, and the relative importance of each component is inferred from the prediction accuracy. In the presence of external forcings, oceanic predictors outperform atmospheric predictors, persistence and random chance baselines out to 25-year leadtimes. Layer-wise relevance propagation is used to uncover the learned sources of predictability, revealing that NNs consistently rely upon the Gulf Stream-North Atlantic Current region for accurate predictions. Additionally, CESM1-trained NNs do not need additional transfer learning to successfully predict the phasing of multidecadal variability in an observational dataset, suggesting consistency in physical processes driving NASST variability between CESM1 and observations.