Effective query-item relevance modeling is pivotal for enhancing user experience and safeguarding user satisfaction in e-commerce search systems. Recently, benefiting from the vast inherent knowledge, Large Language Model (LLM) approach demonstrates strong performance and long-tail generalization ability compared with previous neural-based specialized relevance learning methods. Though promising, current LLM-based methods encounter the following inadequacies in practice: First, the massive parameters and computational demands make it difficult to be deployed online. Second, distilling LLM models to online models is a feasible direction, but the LLM relevance modeling is a black box, and its rich intrinsic knowledge is difficult to extract and apply online. To improve the interpretability of LLM and boost the performance of online relevance models via LLM, we propose an Explainable LLM-driven Multi-dimensional Distillation framework for e-commerce relevance learning, which comprises two core components: (1) An Explainable LLM for relevance modeling (ELLM-rele), which decomposes the relevance learning into intermediate steps and models relevance learning as a Chain-of-Thought (CoT) reasoning, thereby enhancing both interpretability and performance of LLM. (2) A Multi-dimensional Knowledge Distillation (MKD) architecture that transfers the knowledge of ELLM-rele to current deployable interaction-based and representation-based student models from both the relevance score distribution and CoT reasoning aspects. Through distilling the probabilistic and CoT reasoning knowledge, MKD improves both the semantic interaction and long-tail generalization abilities of student models. Extensive offline evaluations and online experiments on Taobao search ad scene demonstrate that our proposed framework significantly enhances e-commerce relevance learning performance and user experience.