The surrogate model is an essential part of modern design optimization and exploration. In some cases, exploration of design space in multi-objective problems is important to reveal useful design insight and guidelines that will be useful for engineers. However, most surrogate models are black boxes, making interpretation difficult. This paper investigates the framework of explainable surrogate models using Shapley Additive Explanations (SHAP) to gain important design insight that helps users better understand the relationship between objective functions and design variables. We applied the explainable surrogate model framework to multi-objective design problems and performed a comparison with active subspaces and Sobol indices. Several techniques to extract design insight based on SHAP values are discussed: the averaged SHAP, the SHAP summary plot, the single- and bi-objective SHAP dependence plot, and the SHAP correlation matrix. Two aerodynamic design cases are selected to demonstrate the capability of explainable surrogate models: nine-variable inviscid and twenty-variable viscous transonic airfoil design. The findings indicate that SHAP provides more valuable insights than active subspaces and Sobol indices, particularly regarding the impact of individual design variables on the objectives. Consequently, SHAP can be employed in conjunction with active subspaces and Sobol indices to explore the input–output relationship in multi-objective design exploration comprehensively.