Large language models (LLMs) have recently achieved remarkable progress in sentence-level machine translation, but scaling to document-level machine translation (DocMT) remains challenging, particularly in modeling long-range dependencies and discourse phenomena across sentences and paragraphs. Document translations generated by LLMs often suffer from poor consistency, weak coherence, and omission errors. To address these issues, we propose SubDocTrans, a novel DocMT framework that enables LLMs to produce high-quality translations through plug-and-play, multi-granularity knowledge extraction and integration. SubDocTrans first performs topic segmentation to divide a document into coherent topic sub-documents. For each sub-document, both global and local knowledge are extracted including bilingual summary, theme, proper nouns, topics, and transition hint. We then incorporate this multi-granularity knowledge into the prompting strategy, to guide LLMs in producing consistent, coherent, and accurate translations. We conduct extensive experiments across various DocMT tasks, and the results demonstrate the effectiveness of our framework, particularly in improving consistency and coherence, reducing omission errors, and mitigating hallucinations.