Radiology report summarization (RRS) is crucial for patient care, requiring concise "Impressions" from detailed "Findings." This paper introduces a novel prompting strategy to enhance RRS by first generating a layperson summary. This approach normalizes key observations and simplifies complex information using non-expert communication techniques inspired by doctor-patient interactions. Combined with few-shot in-context learning, this method improves the model's ability to link general terms to specific findings. We evaluate this approach on the MIMIC-CXR, CheXpert, and MIMIC-III datasets, benchmarking it against 7B/8B parameter state-of-the-art open-source large language models (LLMs) like Meta-Llama-3-8B-Instruct. Our results demonstrate improvements in summarization accuracy and accessibility, particularly in out-of-domain tests, with improvements as high as 5% for some metrics.