Discharge letters (DiL) are used within any hospital Information Systems to track diseases of patients during their hospitalisation. Such records are commonly classified over the standard taxonomy made by the World Health Organization, that is the International Statistical Classification of Diseases and Related Health Problems (ICD-10). Particularly, classifying DiLs on the right code is crucial to allow hospitals to be refunded by Public Administrations on the basis of the health service provided. In many practical cases the classification task is carried out by hospital operators, that often have to cope under pressure, making this task an error-prone and time-consuming activity. This process might be improved by applying machine learning techniques to empower the clinical staff. In this paper, we present a system, namely eXDiL, that uses a two-stage Machine Learning and XAI-based approach for classifying DiL data on the ICD-10 taxonomy. To skim the common cases, we first classify automatically the most frequent codes. The codes that are not automatically discovered will be classified into the appropriate chapter and given to an operator to assess the correct code, in addition to an extensive explanation to help the evaluation, comprising of an explainable local surrogate model and a word similarity task. We also show how our approach will be beneficial to healthcare operators, and in particular how it will speed up the process and potentially reduce human errors.