The Kepler and TESS missions have generated over 100,000 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. Different from the existing machine learning works, ExoMiner, the proposed deep learning classifier in this work, mimics how domain experts examine diagnostic tests to vet a transit signal. ExoMiner is a highly accurate, explainable, and robust classifier that 1) allows us to validate 301 new exoplanets from the MAST Kepler Archive and 2) is general enough to be applied across missions such as the on-going TESS mission. We perform an extensive experimental study to verify that ExoMiner is more reliable and accurate than the existing transit signal classifiers in terms of different classification and ranking metrics. For example, for a fixed precision value of 99%, ExoMiner retrieves 93.6% of all exoplanets in the test set (i.e., recall=0.936) while this rate is 76.3% for the best existing classifier. Furthermore, the modular design of ExoMiner favors its explainability. 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). Accepted for Publication in Astrophysical Journals, November 12, 2021.