Multi-trait automated essay scoring (AES) systems provide a fine-grained evaluation of an essay's diverse aspects. While they excel in scoring, prior systems fail to explain why specific trait scores are assigned. This lack of transparency leaves instructors and learners unconvinced of the AES outputs, hindering their practical use. To address this, we propose a self-explainable Rationale-Driven Multi-trait automated Essay scoring (RaDME) framework. RaDME leverages the reasoning capabilities of large language models (LLMs) by distilling them into a smaller yet effective scorer. This more manageable student model is optimized to sequentially generate a trait score followed by the corresponding rationale, thereby inherently learning to select a more justifiable score by considering the subsequent rationale during training. Our findings indicate that while LLMs underperform in direct AES tasks, they excel in rationale generation when provided with precise numerical scores. Thus, RaDME integrates the superior reasoning capacities of LLMs into the robust scoring accuracy of an optimized smaller model. Extensive experiments demonstrate that RaDME achieves both accurate and adequate reasoning while supporting high-quality multi-trait scoring, significantly enhancing the transparency of AES.