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We introduce the Learning to eXplain Recommendations (LXR) framework, a post-hoc, model-agnostic framework for counterfactual explanations of recommender systems. LXR can work with any differentiable recommender and learns to score the importance of users' personal data with respect to a recommended item. The framework's objective employs a novel self-supervised counterfactual loss term that seeks to spotlight the user data most instrumental in the recommendation of an item. Additionally, we propose several counterfactual evaluation metrics for assessing explanations in recommender systems. Using these metrics, our results demonstrate LXR's capability to provide counterfactual explanations for various recommendation algorithms across different datasets. LXR's code is publicly available at https://github.com/ExplainingRecommendations/LXR.