Corpus distillation for biomedical large language models (LLMs) seeks to address the pressing challenge of insufficient quantity and quality in open-source annotated scientific corpora, which remains a bottleneck for effective LLM training in biomedical research. This paper proposes a knowledge-driven, agentic framework for scientific corpus distillation, tailored explicitly for LLM training in the biomedical domain, addressing the challenge posed by the complex hierarchy of biomedical knowledge. Central to our approach is a collaborative multi-agent architecture, where specialized agents, each guided by the Medical Subject Headings (MeSH) hierarchy, work in concert to autonomously extract, synthesize, and self-evaluate high-quality textual data from vast scientific literature. This agentic framework collectively generates and refines domain-specific question-answer pairs, ensuring comprehensive coverage and consistency with biomedical ontologies while minimizing manual involvement. Extensive experimental results show that language models trained on our multi-agent distilled datasets achieve notable improvements in biomedical question-answering tasks, outperforming both strong life sciences LLM baselines and advanced proprietary models. Notably, our AI-Ready dataset enables Llama3-70B to surpass GPT-4 with MedPrompt and Med-PaLM-2, despite their larger scale. Detailed ablation studies and case analyses further validate the effectiveness and synergy of each agent within the framework, highlighting the potential of multi-agent collaboration in biomedical LLM training.