This paper proposes explainable machine learning models for enhancing gene expression classification. The proposed multi-class 1-norm support vector machine (MC-SVM-1) algorithm adopts the One-Versus-All multi-class strategy, leveraging binary 1-norm SVM models. The inherent sparsity of the 1-norm SVM solution enables automatic suppression of numerous dimensions associated with null weights. This feature elimination significantly enhances classification outcomes for high-dimensional gene expression datasets. Empirical test results on 25 gene expression datasets demonstrate that our MC-SVM-1 algorithm effectively reduces 99% of full dimensions, leading to respective accuracy increases of 7.1% and 4.03% when compared to training SVM and random forest models on the complete gene expression dataset dimensions. Subsequently, principal component analysis and locally interpretable model-agnostic explanations techniques are used to gain insights into how the classification model effectively handles the selected features extracted from gene expression datasets.