Online symptom checkers (OSC) are widely used intelligent systems in health contexts such as primary care, remote healthcare, and epidemic control. OSCs use algorithms such as machine learning to facilitate self-diagnosis and triage based on symptoms input by healthcare consumers. However, intelligent systems' lack of transparency and comprehensibility could lead to unintended consequences such as misleading users, especially in high-stakes areas such as healthcare. In this paper, we attempt to enhance diagnostic transparency by augmenting OSCs with explanations. We first conducted an interview study (N=25) to specify user needs for explanations from users of existing OSCs. Then, we designed a COVID-19 OSC that was enhanced with three types of explanations. Our lab-controlled user study (N=20) found that explanations can significantly improve user experience in multiple aspects. We discuss how explanations are interwoven into conversation flow and present implications for future OSC designs.