The Kepler and TESS missions have generated a large number of potential transit signals that must be processed in order to create a catalog of planet candidates. During the last few years, there has been a growing interest in using machine learning to analyze these data in search of new exoplanets. The existing machine classifiers, however, are not accurate, explainable, or general enough in order to be used in practice to build new catalogs of planet candidates. By mimicking the process by which domain experts examine different types of diagnostic tests to vet a transit signal, we introduce a new classifier, ExoMiner, that can be utilized to mine new exoplanets very effectively. ExoMiner is a highly accurate and explainable deep learning classifier that (1) allows us to validate more than 200 new exoplanets from the MAST Kepler Archive, and (2) is general enough to be applied across missions (e.g. a model trained on Kepler signals can be used to classify TESS signals). We perform an extensive experimental study to verify that ExoMiner is significantly more reliable and accurate than the existing machine-based transit signal classifiers in terms of different classification and ranking evaluation metrics. For example, for a fixed precision value of 0.99, ExoMiner retrieves 93% of all exoplanets in the test set (i.e. recall=0.93) while this rate is 75% for the best existing machine classifier. Besides being accurate and general, the modular design of ExoMiner makes it highly explainable. We introduce a simple explainability framework that provides experts with feedback on why ExoMiner classifies a transit signal into a specific class label (e.g. planet candidate or not planet candidate).