In this paper, we study the explainability of automated data cleaning pipelines and propose CLeanEX, a solution that can generate explanations for the pipelines automatically selected by an automated cleaning system, given it can provide its corresponding cleaning pipeline search space. We propose meaningful explanatory features that are used to describe the pipelines and generate predicate-based explanation rules. We compute quality indicators for these explanations and propose a multi-objective optimization algorithm to select the optimal set of explanations for user-defined objectives. Preliminary experiments show the need for multi-objective optimization for the generation of high-quality explanations that can be either intrinsic to the single selected cleaning pipeline or relative to the other pipelines that were not selected by the automated cleaning system. We also show that CLeanEX is a promising step towards generating automatically insightful explanations, while catering to the needs of the user alike.