Explainable machine learning significantly improves the transparency of deep neural networks~(DNN). However, existing work is constrained to explaining the behavior of individual model predictions, and lacks the ability to transfer the explanation across various models and tasks. This limitation results in explaining various tasks being time- and resource-consuming. To address this problem, we develop a pre-trained, DNN-based, generic explainer on large-scale image datasets, and leverage its transferability to explain various vision models for downstream tasks. In particular, the pre-training of generic explainer focuses on LEarning Transferable Attribution (LETA). The transferable attribution takes advantage of the versatile output of the target backbone encoders to comprehensively encode the essential attribution for explaining various downstream tasks. LETA guides the pre-training of the generic explainer towards the transferable attribution, and introduces a rule-based adaptation of the transferable attribution for explaining downstream tasks, without the need for additional training on downstream data. Theoretical analysis demonstrates that the pre-training of LETA enables minimizing the explanation error bound aligned with the conditional $\mathcal{V}$-information on downstream tasks. Empirical studies involve explaining three different architectures of vision models across three diverse downstream datasets. The experiment results indicate LETA is effective in explaining these tasks without the need for additional training on the data of downstream tasks.