Our motivating application is a real-world problem: COVID-19 classification from CT imaging, for which we present an explainable Deep Learning approach based on a semi-supervised classification pipeline that employs variational autoencoders to extract efficient feature embedding. We have optimized the architecture of two different networks for CT images: (i) a novel conditional variational autoencoder (CVAE) with a specific architecture that integrates the class labels inside the encoder layers and uses side information with shared attention layers for the encoder, which make the most of the contextual clues for representation learning, and (ii) a downstream convolutional neural network for supervised classification using the encoder structure of the CVAE. With the explainable classification results, the proposed diagnosis system is very effective for COVID-19 classification. Based on the promising results obtained qualitatively and quantitatively, we envisage a wide deployment of our developed technique in large-scale clinical studies.Code is available at https://git.etrovub.be/AVSP/ct-based-covid-19-diagnostic-tool.git.