Explanations are hypothesized to improve human understanding of machine learning models and achieve a variety of desirable outcomes, ranging from model debugging to enhancing human decision making. However, empirical studies have found mixed and even negative results. An open question, therefore, is under what conditions explanations can improve human understanding and in what way. To address this question, we first identify three core concepts that cover most existing quantitative measures of understanding: task decision boundary, model decision boundary, and model error. Using adapted causal diagrams, we provide a formal characterization of the relationship between these concepts and human approximations (i.e., understanding) of them. The relationship varies by the level of human intuition in different task types, such as emulation and discovery, which are often ignored when building or evaluating explanation methods. Our key result is that human intuitions are necessary for generating and evaluating machine explanations in human-AI decision making: without assumptions about human intuitions, explanations may improve human understanding of model decision boundary, but cannot improve human understanding of task decision boundary or model error. To validate our theoretical claims, we conduct human subject studies to show the importance of human intuitions. Together with our theoretical contributions, we provide a new paradigm for designing behavioral studies towards a rigorous view of the role of machine explanations across different tasks of human-AI decision making.