Digital Twin (DT) systems are increasingly applied in environmental contexts to support real-time monitoring, forecasting, and decision-making. While technically advanced, many DT implementations fall short in communicating insights in ways that are accessible, interpretable, and actionable for diverse audiences. This paper introduces a user-centered framework that enhances the communicative power of environmental DTs through the integration of data storytelling techniques. The proposed approach is structured around five components: (i) a four-phase data storytelling process, (ii) a user-system interaction model driven by user intent, (iii) progressive levels of knowledge formalization, (iv) a workflow that adapts key data storytelling activities to generate DT outputs that are both analytically robust and communication-ready, and (v) a semantic mapping table that links analytical operations to narrative and visual outputs. Together, these components enable DT systems to transform analytical results into meaningful, audience-adapted stories. The framework is illustrated through an environmental prediction use case, demonstrating its potential to enhance interpretability, increase user engagement, and support more informed decision-making.