Breast cancer is a type of cancer caused by the uncontrolled growth of cells in the breast tissue. In a few cases, erroneous diagnosis of breast cancer by specialists and unnecessary biopsies can lead to various negative consequences. In some cases, radiologic examinations or clinical findings may raise the suspicion of breast cancer, but subsequent detailed evaluations may not confirm cancer. In addition to causing unnecessary anxiety and stress to patients, such diagnosis can also lead to unnecessary biopsy procedures, which are painful, expensive, and prone to misdiagnosis. Therefore, there is a need for the development of more accurate and reliable methods for breast cancer diagnosis. In this study, we proposed an artificial intelligence (AI)-based method for automatically classifying breast solid mass lesions as benign vs malignant. In this study, a new breast cancer dataset (Breast-XD) was created with 791 solid mass lesions belonging to 752 different patients aged 18 to 85 years, which were examined by experienced radiologists between 2017 and 2022. Six classifiers, support vector machine (SVM), K-nearest neighbor (K-NN), random forest (RF), decision tree (DT), logistic regression (LR), and XGBoost, were trained on the training samples of the Breast-XD dataset. Then, each classifier made predictions on 159 test data that it had not seen before. The highest classification result was obtained using the explainable XGBoost model (X2GAI) with an accuracy of 94.34%. An explainable structure is also implemented to build the reliability of the developed model. The results obtained by radiologists and the X2GAI model were compared according to the diagnosis obtained from the biopsy. It was observed that our developed model performed well in cases where experienced radiologists gave false positive results.