This work addresses the challenges of robust covariance estimation and interpretable outlier detection for multivariate functional data with separable covariance structures. Our goal is to develop a method that simultaneously improves robustness and interpretability in this context. We establish a connection between stochastic processes with separable covariance structures and the corresponding matrix-variate distribution of their basis representations. Leveraging this connection, we employ the Matrix Minimum Covariance Determinant (MMCD) approach introduced by [Mayrhofer et al., 2024], in conjunction with a multivariate functional Mahalanobis (semi-)distance introduced in [Galeano et al., 2015], to robustly estimate both mean and covariance functions for multivariate functional data. For interpretable outlier detection, we propose a methodology that applies Shapley values from game theory to decompose overall outlyingness into component-specific contributions. Importantly, we reduce the otherwise exponential computational complexity (relative to the number of components) to linear complexity, while retaining the key properties of the Shapley value. This integrated framework—combining robust Mahalanobis distances, MMCD estimators, and Shapley value-based outlyingness decomposition provides a robust and interpretable approach for analyzing multivariate functional data with separable covariance structures. The effectiveness of this approach is demonstrated through both theoretical analysis and practical applications, including simulations and real-world case studies.