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Deep learning applied to magnetic resonance imaging data have shown great promise as a translational technology for diagnosis and prognosis in dementia, but its impact clinically has thus far been limited. This is partially attributed to the opaqueness of deep learning models, causing insufficient understanding of what underlies their decisions. To overcome this, we trained convolutional neural nets to differentiate patients with dementia from healthy controls, and applied layerwise relevance propagation to procure individual-level explanations of the model predictions. Through extensive validations we demonstrate that patterns recognized by the model corroborate existing knowledge of neuropathology in dementia. Then, employing the explainable dementia classifier in a longitudinal dataset of patients with mild cognitive impairment, we show that the spatially rich explanations complement the prediction for prognosis, and help characterize the personalized manifestation of disease. Overall, our work exemplifies the clinical potential of explainable artificial intelligence in precision medicine.