Sea Surface Temperature (SST) is crucial for understanding upper-ocean thermal dynamics and ocean-atmosphere interactions, which have profound economic and social impacts. While data-driven models show promise in SST prediction, their black-box nature often limits interpretability and overlooks key physical processes. Recently, physics-informed neural networks have been gaining momentum but struggle with complex ocean-atmosphere dynamics due to 1) inadequate characterization of seawater movement (e.g., coastal upwelling) and 2) insufficient integration of external SST drivers (e.g., turbulent heat fluxes). To address these challenges, we propose SSTODE, a physics-informed Neural Ordinary Differential Equations (Neural ODEs) framework for SST prediction. First, we derive ODEs from fluid transport principles, incorporating both advection and diffusion to model ocean spatiotemporal dynamics. Through variational optimization, we recover a latent velocity field that explicitly governs the temporal dynamics of SST. Building upon ODE, we introduce an Energy Exchanges Integrator (EEI)-inspired by ocean heat budget equations-to account for external forcing factors. Thus, the variations in the components of these factors provide deeper insights into SST dynamics. Extensive experiments demonstrate that SSTODE achieves state-of-the-art performances in global and regional SST forecasting benchmarks. Furthermore, SSTODE visually reveals the impact of advection dynamics, thermal diffusion patterns, and diurnal heating-cooling cycles on SST evolution. These findings demonstrate the model's interpretability and physical consistency.