Classification of complex and large hyperspectral images (HSIs) with machine learning (ML) algorithms is an important research area. Recently, explainable artificial intelligence (XAI), which helps to explain and interpret black-box ML algorithms, has become popular. Our study aims to present extensive research on the use of XAI methods in explaining the band effect in HSI classification and the impact of reducing the high band number of HSIs by feature selection on the performance of the classifiers. The importance levels of the spectral bands that are effective in the decisions of the different ML classifiers were examined with the deep reinforcement learning and XAI methods, such as Shapley additive explanations (SHAP) and permutation feature importance (PFI). Our work selects representative bands using SHAP and PFI as XAI analysis techniques. We evaluated the XAI-based band selection performance on three publicly available HSI datasets using random forest, light gradient-boosting machine, and extreme gradient boosting classifier algorithms. The results obtained by applying XAI and deep learning methods were used to select spectral bands. Additionally, principal component analysis, a common dimension reduction technique, was performed on the dataset used in our study. Comparable performance evaluation shows that XAI-based methods choose informative bands and outperform other methods in the subsequent tasks. Thus the global and class-based effects of spectral bands can be explained, and the performance of classifiers can be improved by eliminating features that have a negative impact on classification. In HSI classification, studies examining the decisions of ML classifiers using XAI techniques are limited. Our study is one of the pioneer studies in the usage of XAI in HSI classification.