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Mistrust, amplified by numerous artificial intelligence (AI) related incidents, is an issue that has caused the energy and industrial sectors to be amongst the slowest adopter of AI methods. Central to this issue is the black-box problem of AI, which impedes investments and is fast becoming a legal hazard for users. Explainable AI (XAI) is a recent paradigm to tackle such an issue. Being the backbone of the industry, the prognostic and health management (PHM) domain has recently been introduced into XAI. However, many deficiencies, particularly the lack of explanation assessment methods and uncertainty quantification, plague this young domain. In the present paper, we elaborate a framework on explainable anomaly detection and failure prognostic employing a Bayesian deep learning model and Shapley additive explanations (SHAP) to generate local and global explanations from the PHM tasks. An uncertainty measure of the Bayesian model is utilized as a marker for anomalies and expands the prognostic explanation scope to include the model's confidence. In addition, the global explanation is used to improve prognostic performance, an aspect neglected from the handful of studies on PHM-XAI. The quality of the explanation is examined employing local accuracy and consistency properties. The elaborated framework is tested on real-world gas turbine anomalies and synthetic turbofan failure prediction data. Seven out of eight of the tested anomalies were successfully identified. Additionally, the prognostic outcome showed a 19% improvement in statistical terms and achieved the highest prognostic score amongst best published results on the topic. [ABSTRACT FROM AUTHOR]