Precision medicine requires understanding the underlying drivers of heterogeneous treatment responses. Although machine learning methods have shown promise for estimating patient-specific treatment effects, their clinical utility remains limited because they often function as ``black box'' predictors that fail to explain why responses vary across individuals. Here we present ALEX, an explainable AI (XAI)-driven, multi-agent framework that addresses this interpretability gap by translating the patient variables driving these predictions into data-grounded, natural-language clinical explanations. ALEX first performs XAI analysis on treatment effect estimation and couples the intermediate results with large language model (LLM) agents to produce contextualized clinical insights. Across five landmark randomized controlled trials, ALEX outperformed existing agentic methods on explanation quality metrics and alignment with the biomedical literature. In empirical case studies, ALEX identified baseline glucose level as a potential explanation for the divergent findings between the ACCORD-BP and SPRINT trials, and proposed age as a key effect modifier for pre-hospital tranexamic acid efficacy. These findings suggest that ALEX can help translate treatment effect heterogeneity into clinically grounded explanations for further investigation.