Knee osteoarthritis (OA) is the most common musculoskeletal disorder. OA diagnosis is currently conducted by assessing symptoms and evaluating plain radiographs, but this process suffers from subjectivity. In this study, we present a new transparent computer-aided diagnosis method based on the Deep Siamese Convolutional Neural Network to automatically score knee OA severity according to the Kellgren-Lawrence grading scale. We trained our method using the data solely from the Multicenter Osteoarthritis Study and validated it on randomly selected 3,000 subjects (5,960 knees) from Osteoarthritis Initiative dataset. Our method yielded a quadratic Kappa coefficient of 0.83 and average multiclass accuracy of 66.71% compared to the annotations given by a committee of clinical experts. Here, we also report a radiological OA diagnosis area under the ROC curve of 0.93. Besides this, we present attention maps highlighting the radiological features affecting the network decision. Such information makes the decision process transparent for the practitioner, which builds better trust toward automatic methods. We believe that our model is useful for clinical decision making and for OA research; therefore, we openly release our training codes and the data set created in this study.