Explainable Artificial Intelligence (XAI) has recently gained a swell of interest, as many Artificial Intelligence (AI) practitioners and developers are compelled to rationalize how such AI-based systems work. Decades back, most XAI systems were developed as knowledge-based or expert systems. These systems assumed reasoning for the technical description of an explanation, with little regard for the user's cognitive capabilities. The emphasis of XAI research appears to have turned to a more pragmatic explanation approach for better understanding. An extensive area where cognitive science research may substantially influence XAI advancements is evaluating user knowledge and feedback, which are essential for XAI system evaluation. To this end, we propose a framework to experiment with generating and evaluating the explanations on the grounds of different cognitive levels of understanding. In this regard, we adopt Bloom's taxonomy, a widely accepted model for assessing the user's cognitive capability. We utilize the counterfactual explanations as an explanation-providing medium encompassed with user feedback to validate the levels of understanding about the explanation at each cognitive level and improvise the explanation generation methods accordingly.